How to apply conditional probability - AP Statistics

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Question

How many different football teams of 11 members may be created from 13 men, without regard to the position played by any of the members?

Answer

There are 13 men to choose from in order to make an 11 member football team. Thus, the answer is set up as: 13 choose 11: $\frac{13!}{11!2!}=78$ .

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Question

Number of Emails
Probability

The number of emails that Sam can respond to per hour is given by this probability distribution. What is the average number of emails that Sam respons to per hour?

Answer

The answer is found by multiplying by the probability :

emails

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Question

A child has a bag of marbles-- red, blue, and yellow. The child randomly selects one marble and gives it away. The child then selects a second marble. What is the probability that both marbles selected were yellow?

Answer

The two events in this case are dependent (the result of the first event will affect the outcome probabilities of the second event). Find the probability of a particular outcome for both events by multiplying the probability of event A by the probability of event B given the result of A. In other words, multiply the probability of both events, remembering to account for the result of event A in determining the probability of event B. Here, the probability of selecting a yellow marble the first time is 3/8, and the second time is 2/7.

$3/8\cdot 2/7=6/56$

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Question

There are 10 horses in a herd. 4 are males, 6 are females. 7 of the horses are brown while the remaining 3 are white. 5 female horses are brown. A horse is randomly selected from the herd. Given that the horse is brown, what is the probability that the horse is a male?

Answer

The probability of any event occurring is the number of outcomes in which case that event occurs divided by the number of total number of outcomes. Here you know that the total number of brown horses is 7, and if 5 of those brown horses are female then 2 must be male. So the probability that a brown horse is male is 2/7.

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Question

There are 10 horses in a herd. 4 are males, 6 are females. 7 of the horses are brown while the remaining 3 are white. At least 1 male horse is white and 5 female horses are brown.

One horse is selected at random from the herd. What is the probability of selecting a female horse given that the horse selected is white?

Answer

This question asks for the probability of an event given that one event has occurred, meaning it is a conditional probability. The equation for a conditional probability is:

$P(A|B)=\frac{P(A\cap B)}{P(A)}$

It is important to properly assign the A and B variables according to the question. It helps to read the equation out loud: "The probability of A given B equals the probability of A and B divided by the probability of A."

We need to find the probability of a female given that the horse is white. Therefore, female horse is A, white horse is B.

So, find each probability needed for the equation.

$P(A)=P(female)=\frac{6}{10}=0.6$

$P(B)=P(white)=\frac{3}{10}=0.3$

If 6 of the 7 females are brown, then 1 horse out of 10 is a white female horse.

$P(A\cap B)= P(female\cap white)= \frac{2}{10}=0.2$

Now, use the information to solve the equation.

$P(A|B)= \frac{0.2}{0.6}=0.33$

Alternatively you could simply count out the number of favorable outcomes: there are 6 total female horses and 5 are brown, so that leaves 1 white female horse (the "favorable outcome" here). And there are 3 total white horses. So if you know that you're selecting from the pool of 3 white horses and only 1 is female, the probability of choosing a female white horse from that set is 1/3.

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Question

There are 10 horses in a herd. 4 are males, 6 are females. 7 of the horses are brown while the remaining 3 are white. 1 male horse is white, 5 female horses are brown.

A horse is randomly selected from the herd. What is the probability that the horse chosen is brown or female?

Answer

This problem asks for the probability of one event or another, so the addition rule of probability is approriate. However, brown horse and female horse are not mutually exclusive events, because there are brown female horses. Therefore we must subtract the brown male horses to avoid double counting.

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

$P(brown)=\frac{7}{10}$

$P(female)=\frac{6}{10}$

$P(female\cap brown)=\frac{5}{10}$

Now, fill in the equation.

$P(female \cup brown)=\frac{6}{10}+\frac{7}{10}-\frac{5}{10}=\frac{8}{10}=\frac{4}{5}$

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Question

The probability that there will be an accident on Highway each day is determined by the weather. If it's a dry day, there is a % chance of an accident, and a % chance if it's a wet day. The radio says that there is a % chance of rain today. What is the probability that there will be an accident on Highway today?

Answer

The chance of an accident is dependent on the weather conditions, so that is the first branch on the tree diagram.

The probabilities at the next branch are conditional and the last two branches have the chance of accident.

$P(wet \cap accident) = 0.2 \cdot0.010 = 0.0020$ and $P(dry\cap accident) = 0.80 \cdot 0.002 = 0.0016$ ,

so the total probability of an accident is .

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Question

There is a sample of 100 randomly selected children (60 of which are boys and 40 are girls). They were asked if they liked to play football. 55 boys answered yes to the question and 5 of the girls aswered yes to the question. What is the probability that a child said yes to the question give that they are a boy?

Answer

The conditional probability formula is the probability of both events happening divided by the probability of the given event. The probability of saying "yes" and also being a boy is $\frac{55}{100}$ . The probability of being a boy in the sample is $\frac{60}{100}$ . So $\frac{55}{100}\div\frac{60}{100}$ = $\frac{55}{60}$ = .

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Question

How many different football teams of 11 members may be created from 13 men, without regard to the position played by any of the members?

Answer

There are 13 men to choose from in order to make an 11 member football team. Thus, the answer is set up as: 13 choose 11: $\frac{13!}{11!2!}=78$ .

Compare your answer with the correct one above

Question

Number of Emails
Probability

The number of emails that Sam can respond to per hour is given by this probability distribution. What is the average number of emails that Sam respons to per hour?

Answer

The answer is found by multiplying by the probability :

emails

Compare your answer with the correct one above

Question

A child has a bag of marbles-- red, blue, and yellow. The child randomly selects one marble and gives it away. The child then selects a second marble. What is the probability that both marbles selected were yellow?

Answer

The two events in this case are dependent (the result of the first event will affect the outcome probabilities of the second event). Find the probability of a particular outcome for both events by multiplying the probability of event A by the probability of event B given the result of A. In other words, multiply the probability of both events, remembering to account for the result of event A in determining the probability of event B. Here, the probability of selecting a yellow marble the first time is 3/8, and the second time is 2/7.

$3/8\cdot 2/7=6/56$

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Question

There are 10 horses in a herd. 4 are males, 6 are females. 7 of the horses are brown while the remaining 3 are white. 5 female horses are brown. A horse is randomly selected from the herd. Given that the horse is brown, what is the probability that the horse is a male?

Answer

The probability of any event occurring is the number of outcomes in which case that event occurs divided by the number of total number of outcomes. Here you know that the total number of brown horses is 7, and if 5 of those brown horses are female then 2 must be male. So the probability that a brown horse is male is 2/7.

Compare your answer with the correct one above

Question

There are 10 horses in a herd. 4 are males, 6 are females. 7 of the horses are brown while the remaining 3 are white. At least 1 male horse is white and 5 female horses are brown.

One horse is selected at random from the herd. What is the probability of selecting a female horse given that the horse selected is white?

Answer

This question asks for the probability of an event given that one event has occurred, meaning it is a conditional probability. The equation for a conditional probability is:

$P(A|B)=\frac{P(A\cap B)}{P(A)}$

It is important to properly assign the A and B variables according to the question. It helps to read the equation out loud: "The probability of A given B equals the probability of A and B divided by the probability of A."

We need to find the probability of a female given that the horse is white. Therefore, female horse is A, white horse is B.

So, find each probability needed for the equation.

$P(A)=P(female)=\frac{6}{10}=0.6$

$P(B)=P(white)=\frac{3}{10}=0.3$

If 6 of the 7 females are brown, then 1 horse out of 10 is a white female horse.

$P(A\cap B)= P(female\cap white)= \frac{2}{10}=0.2$

Now, use the information to solve the equation.

$P(A|B)= \frac{0.2}{0.6}=0.33$

Alternatively you could simply count out the number of favorable outcomes: there are 6 total female horses and 5 are brown, so that leaves 1 white female horse (the "favorable outcome" here). And there are 3 total white horses. So if you know that you're selecting from the pool of 3 white horses and only 1 is female, the probability of choosing a female white horse from that set is 1/3.

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Question

There are 10 horses in a herd. 4 are males, 6 are females. 7 of the horses are brown while the remaining 3 are white. 1 male horse is white, 5 female horses are brown.

A horse is randomly selected from the herd. What is the probability that the horse chosen is brown or female?

Answer

This problem asks for the probability of one event or another, so the addition rule of probability is approriate. However, brown horse and female horse are not mutually exclusive events, because there are brown female horses. Therefore we must subtract the brown male horses to avoid double counting.

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

$P(brown)=\frac{7}{10}$

$P(female)=\frac{6}{10}$

$P(female\cap brown)=\frac{5}{10}$

Now, fill in the equation.

$P(female \cup brown)=\frac{6}{10}+\frac{7}{10}-\frac{5}{10}=\frac{8}{10}=\frac{4}{5}$

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Question

The probability that there will be an accident on Highway each day is determined by the weather. If it's a dry day, there is a % chance of an accident, and a % chance if it's a wet day. The radio says that there is a % chance of rain today. What is the probability that there will be an accident on Highway today?

Answer

The chance of an accident is dependent on the weather conditions, so that is the first branch on the tree diagram.

The probabilities at the next branch are conditional and the last two branches have the chance of accident.

$P(wet \cap accident) = 0.2 \cdot0.010 = 0.0020$ and $P(dry\cap accident) = 0.80 \cdot 0.002 = 0.0016$ ,

so the total probability of an accident is .

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Question

There is a sample of 100 randomly selected children (60 of which are boys and 40 are girls). They were asked if they liked to play football. 55 boys answered yes to the question and 5 of the girls aswered yes to the question. What is the probability that a child said yes to the question give that they are a boy?

Answer

The conditional probability formula is the probability of both events happening divided by the probability of the given event. The probability of saying "yes" and also being a boy is $\frac{55}{100}$ . The probability of being a boy in the sample is $\frac{60}{100}$ . So $\frac{55}{100}\div\frac{60}{100}$ = $\frac{55}{60}$ = .

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Question

How many different football teams of 11 members may be created from 13 men, without regard to the position played by any of the members?

Answer

There are 13 men to choose from in order to make an 11 member football team. Thus, the answer is set up as: 13 choose 11: $\frac{13!}{11!2!}=78$ .

Compare your answer with the correct one above

Question

Number of Emails
Probability

The number of emails that Sam can respond to per hour is given by this probability distribution. What is the average number of emails that Sam respons to per hour?

Answer

The answer is found by multiplying by the probability :

emails

Compare your answer with the correct one above

Question

A child has a bag of marbles-- red, blue, and yellow. The child randomly selects one marble and gives it away. The child then selects a second marble. What is the probability that both marbles selected were yellow?

Answer

The two events in this case are dependent (the result of the first event will affect the outcome probabilities of the second event). Find the probability of a particular outcome for both events by multiplying the probability of event A by the probability of event B given the result of A. In other words, multiply the probability of both events, remembering to account for the result of event A in determining the probability of event B. Here, the probability of selecting a yellow marble the first time is 3/8, and the second time is 2/7.

$3/8\cdot 2/7=6/56$

Compare your answer with the correct one above

Question

There are 10 horses in a herd. 4 are males, 6 are females. 7 of the horses are brown while the remaining 3 are white. 5 female horses are brown. A horse is randomly selected from the herd. Given that the horse is brown, what is the probability that the horse is a male?

Answer

The probability of any event occurring is the number of outcomes in which case that event occurs divided by the number of total number of outcomes. Here you know that the total number of brown horses is 7, and if 5 of those brown horses are female then 2 must be male. So the probability that a brown horse is male is 2/7.

Compare your answer with the correct one above

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