Independent and Dependent Events - AP Statistics

Card 0 of 80

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

What is the probability of getting a sum of when rolling two six-sided fair dice?

Answer

The sample space, or total possible outcomes, when rolling two six-sided dice is .

Ways to get what you want:

So there are ways to get a .

So the probability becomes $\frac{15}{36}=\frac{5}{12}$

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Question

Mary randomly selects the king of hearts from a deck of cards. She then replaces the card and again selects one card from the deck. The selection of the second card is a(n)_________ event.

Answer

The selection of the second card is an independent event because it is unaffected by the first event. If the king of hearts had not been replaced, then the probability of selecting a particular card would have been affected by the first event, and the second selection would have been dependent. This, however, is not the case in this question.

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Question

Given a pair of fair dice, what is the probability of rolling a 7 in one throw?

Answer

There are 36 total outcomes for this experiment and there are six ways to roll a 7 with two dice: 1,6; 6,1; 2,5; 5,2; 3,4; and 4,3. Thus, 6/36 = 1/6.

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Question

A fair coin is tossed into the air a total of ten times and the result, heads or tails, is the face landing up. What is the total number of possible outcomes for this experiment?

Answer

There are two outcomes in each trial of this experiment, and there are ten total trials. Thus, 2 raised to the tenth power yields an answer of 1024.

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Question

Each answer choice describes two events. Which of the following describes independent events?

Answer

Two events are independent of each other when the result of one does not affect the result of the other. In the case of the coin being flipped, the first result in no way influenced the result of the second coin flip. In contrast, when a card is removed from a deck of cards and set aside, that card cannot be selected when a second card is taken from the deck.

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Question

Events and are known to be independent.

while $P(A \bigcap B) = 0.4$ . What must be?

Answer

Because the two events are known to be independent, then the following is true by definition.

$P(A \bigcap B) = P(A)\cdot P(B)$ .

This then becomes an algebra problem:

$P(A \bigcap B) = P(A)\cdot P(B)$

$0.4 = 0.5\cdot P(B)$

$P(B) = \frac{0.4}{0.5} = 0.8$

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Question

Two events and are independent, and while . What is $P(A^C \bigcap B^C)$ ?

Answer

Because the two are independent, the calculation becomes the product of the two by definition.

We need to recall that respresents the compliment of A which is everything that is not in A or in mathematical terms:

.

Likewise for the compliment of B:

Therefore to find the intersection of these two independent events we multiply them together.

$P(A^C \bigcap B^C) = [1 - P(A)][1 - P(B)]$

$= (1 - 0.8)(1 - 0.9) = 0.2\cdot 0.1 = 0.02$

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Question

True or false: When drawing two cards with replacement, the event drawing a spade first is independent of the event drawing a heart second.

Answer

These two events are independent of one another. During sampling with replacement, the first card does not affect the second card being picked.

To illustrate, consider the probability of drawing a heart first

$P(Heart)=\frac{13}{52}$

Assuming you first drew a heart and replaced it in the deck, does the probability of drawing a heart as the second card change?

$P(Heart\left | Spade )=\frac{13}{52}$

The probability remains the same, there are still 13 hearts and 52 total cards.

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Question

True or False: When 2 cards are drawn without replacement from a regular deck of 52 cards, the event of drawing a heart first independent of the event of drawing a heart second.

Answer

These events are not independent, because if one event happens, it affects the probability of the other event happening. Consider the probability of drawing a heart and the probability of getting a heart given a heart was already drawn. If these two probabilities are the same, the events are independent. If the two probabilities are not the asme, the events are not independent.

$P(Heart)=\frac{13}{52}$

$P(Heart | Heart )=\frac{12}{51}$

After a heart has already been drawn, there are now only 52 cards total and 12 hearts left. These two probabilities are not equal, therefore the events are not independent.

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Question

True or false: Two events which are mutually exclsuve are also independent events.

Answer

Events which are mutually exclusive are dependent events. This is becauce if one event happens, it affects the probability that the other event will happen.

Example- Tonight you may attend a soccer game or a basketball game. These events are mutually exclusive because they share no overlap. However, if you go to the basketball game, it changes the probability of you going to the soccer game to zero. Therefore, mutually exclusive events are not independent.

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Question

True or false:

A family has 3 boys. The probability that the fourth child will also be a boy is less than 50%

Answer

The gender of each child can be considered an independent event. Each child has a 50% chance of being a boy, and whether a boy was already born previously does not affect the next child's gender.

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Question

A bag is set in front of you with three different colored marbles inside. There are 9 red marbles, 10 blue marbles, and 7 yellow marbles. What is the probability that you draw a marble that is not yellow?

Answer

First you find the total amount of marbles that are in the bag which is 26 (). Then you must add up all of the marbles that are not yellow which is 19 (9 red marbles + 10 blue marbles). Finally, you must divide the total marbles that are apart of you set (19) by the total marbles (26) and that gives you a probability of .

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