All SAT Math Resources
Example Questions
Example Question #91 : How To Find The Probability Of An Outcome
We randomly pick two numbers from positive integers. What is the probability that their sum is odd given that their product is even?
There are four combinations total, being: {(odd,odd), (odd, even), (even, odd), (even, even)}
Given that their product is even, we only have the set {(odd,even),(even,odd),(even, even)}
The probability that their sum is odd from the set {(odd,even),(even,odd),(even, even)} is because and .
Example Question #3172 : Sat Mathematics
Bill rolls three 6-sided dice at once. The first die comes up as a 5. What is the probability that the total of all three dice will not be a prime number?
The first die comes up as a 5. We are essentially asking about outcomes of two dice, plus 5. As a first step, we should be sure we know the primes we need to exempt. The smallest sum we can have is if both of the other dice roll 1, giving us a 7. 7 is a prime number, so will will exempt this number. The largest number we can roll is both rolling 6, giving us a total of 17 (also a prime). All the primes we will need to remove are then: 7,11,13 and 17.
We have two approaches now. We can find the odds of rolling a number besides those above directly, or we can take the odds of getting any of the above 4, and then subtracting this value from 1.
The latter method is faster, but both use the same method. Firstly, we'll subtract 5 from all the numbers: 2,6,8,12.
The odds of getting a 2 on two dice is , since the only roll that will work is two 1s and there are a total of 36 possible rolls (6 choices per die).
Similarly, the odds of getting a 12 is since it requires two 6's.
For 6, you can roll 1-5, 2-4, 3-3, 4-2, 5-1. This gives us odds of getting a 6. Similarly, an 8 can be achieved by rolling 6-2, 5-3, 4-4, 3-5, 2-6.
Our total odds of rolling any of these 4 is thus:
Our answer is then:
Example Question #3173 : Sat Mathematics
The first three songs of The Silver Comet’s demo album are 2 minutes 30 seconds, 3 minutes 20 seconds, and 3 minutes 45 seconds in length. Jane has the three songs on a continuous loop all night long. What is the probability that her dad tells her to go to sleep during the first song?
2:30 = 150 seconds
3:20 = 200 seconds
3:45 = 225 seconds
Example Question #3174 : Sat Mathematics
All of Jean's brothers have red hair.
If the statement above is true, then which of the following CANNOT be true?
If Winston does not have red hair, then he is not Jean's brother.
If George does not have red hair, then he is Jean's brother.
If Ron has red hair, then he is Jean's brother.
If Eddie is Jean's brother, then he does not have red hair.
If Paul is not Jean's brother, then he has red hair.
If Eddie is Jean's brother, then he does not have red hair.
Here we have a logic statement:
If A (Jean's brother), then B (red hair).
"If Ron has red hair, then he is Jean's brother" states "If B, then A" - we do not know whether or not this is true.
"If Winston does not have red hair, then he is not Jean's brother" states "If not B, then not A" - this has to be true.
"If Paul is not Jean's brother, then he has red hair" states "If not A, then B." We do not know whether or not this is true.
"If Eddie is Jean's brother, then he does not have red hair" states "If A, then not B" - we know this cannot be true.
Example Question #1821 : Problem Solving Questions
An auto insurer underwrites its 60 customers and classifies them in 3 mutually exclusive risk classes. 15 of the customers are in the high-risk class, 35 are in the moderate-risk class, and 10 are in the low-risk class. What is the probability that a randomly selected customer will be in the moderate- or high-risk class?
The probability that a randomly selected customer is in the moderate- or high-risk class is simply the sum of the number of clients in the moderate-risk class and the number of clients in the high-risk class divided by the total number of clients:
This is also equivalent to just summing up the probability of being in the high-risk class and the probability of being in the moderate-risk class. Since the 3 classes are mutually exclusive, we do not have to worry about subtracting the probability of mutual elements.
Example Question #91 : Outcomes
Ben only goes to the park when it is sunny.
If the above statement is true, which of the following is also true?
If Ben is not at the park, it is not sunny.
If it is sunny, Ben is at the park.
If it is not sunny, Ben is not at the park.
If it is rainy, Ben is at the park.
If it is not sunny, Ben is not at the park.
“Ben only goes to the park when it is sunny.” This means it has to be sunny out for Ben to go to the park, but it does not mean that he always is at the park when it is sunny. Looking at the first choice we can say that this is not necessarily true because it could be sunny and Ben doesn’t have to be at the park. Similar reasoning would prove the second choice wrong as well. The third choice is correct-Ben only is at the park when it is sunny, so he’s definitely not there when it’s not sunny. The fourth choice is clearly wrong because we know Ben only goes when it is sunny, not when it’s rainy.
Example Question #101 : How To Find The Probability Of An Outcome
Presented with a deck of fifty-two cards (no jokers), what is the probability of drawing either a face card or a spade?
A face card constitutes a Jack, Queen, or King, and there are twelve in a deck, so the probability of drawing a face card is .
There are thirteen spades in the deck, so the probability of drawing a spade is .
Keep in mind that there are also three cards that fit into both categories: the Jack, Queen, and King of Spades; the probability of drawing one is
Thus the probability of drawing a face card or a spade is:
Example Question #44 : Probability
A coin is flipped four times. What is the probability of getting heads at least three times?
Since this problem deals with a probability with two potential outcomes, it is a binomial distribution, and so the probability of an event is given as:
Where is the number of events, is the number of "successes" (in this case, a "heads" outcome), and is the probability of success (in this case, fifty percent).
Per the question, we're looking for the probability of at least three heads; three head flips or four head flips would satisfy this:
Thus the probability of three or more flips is:
Example Question #41 : Probability
Rolling a four-sided dice, what is the probability of rolling a three times out of four?
Since this problem deals with a probability with two potential outcomes, it is a binomial distribution, and so the probability of an event is given as:
Where is the number of events, is the number of "successes" (in this case rolling a four), and is the probability of success (one in four).
Example Question #41 : How To Find The Probability Of An Outcome
A coin is flipped seven times. What is the probability of getting heads six or fewer times?
Since this problem deals with a probability with two potential outcomes, it is a binomial distribution, and so the probability of an event is given as:
Where is the number of events, is the number of "successes" (in this case, a "heads" outcome), and is the probability of success (in this case, fifty percent).
One approach is to calculate the probability of flipping no heads, one head, two heads, etc., all the way to six heads, and adding those probabilities together, but that would be time consuming. Rather, calculate the probability of flipping seven heads. The complement to that would then be the sum of all other flip probabilities, which is what the problem calls for:
Therefore, the probability of six or fewer heads is: