SE - Math - 05

What is the probability of rolling a prime number on a 6-sided die?

• 1/2

• 1/6

• 2/3

• 5/6

• 0

Answer explanation

A prime number is any number whose prime factorization is only itself and 1. 1 is not counted as a prime number (mathematicians call it that “unique” number, neither prime nor composite). 2, 3, and 5 are prime, while 4 and 6 are composite, so 3 of the 6 faces are prime. This gives us 3 possible ways to roll a prime out of 6 possible outcomes, so the probability is 3/6, or 1/2.

If there are 3 blue marbles, 6 red marbles, 1 yellow marble, and 3 white marbles in a bag, what is the probability of drawing a blue marble?

• 10/13

• 3/10

• 1/2

• 3/13

• 1/3

Answer explanation

There are 13 total marbles in the bag (3 + 6 + 1 + 3 = 13), and 3 of them are blue, so there are 3 ways to draw a blue marble out of 13 possible draws. The probability of drawing a blue marble is then 3/13.

A bag contains only red marbles and blue marbles. If the probability of drawing a blue marble is 0.3, what is the probability of drawing a red marble?

• 0.3

• 0.7

• 0.6

• 0.5

• 0.4

Answer explanation

While we don’t know how many marbles there are, we know that the probability of drawing EITHER a red or a blue marble is 100%, because those are the only colors in the bag. Therefore, 𝑝(drawing a blue marble) + 𝑝(drawing a red marble) = 1. We know that 𝑝(drawing a blue marble) is 0.3, so the equation for 𝑝(drawing a red marble) is 0.3 +𝑝(drawing a red marble) = 1 Therefore the probability of drawing a red marble is 1 − 0.3 = 0.7.

Using 2 6-sided dice, what is the probability that the numbers rolled add to 6?

• 1/36

• 1/6

• 5/36

• 1

• 0

Answer explanation

Sally has a bad habit of sleeping in. She sleeps through her first class of the day 15% of the time, and she sleeps through her second class of the day 7% of the time. If Sally wakes up in time, she always attends every class she isn’t late for. If Sally is seen in her second class, what is the probability that she slept through her first class?

• 8.6%

• 15%

• 8%

• 1.05%

• 46.7%

Answer explanation

First, notice that if Sally isn’t in her second class, then she definitely slept through her first class. Also notice that if Sally is in her first class, then she is definitely in her second class. Therefore, the 7% of times that Sally sleeps through her second class is the same as the percentage of times that Sally sleeps through both her first and second classes. We know then that 15%−7%=8% of the time, Sally is in her second class but not her first class, and the other 85% of the time she is in both classes. Thus out of 93% of days where Sally is present for her second class, Sally missed her first class (8/93)% of the time, which gives us 8.6% probability that Sally slept through her first class but not her second class.

While flipping a fair coin, you notice that the last 4 flips have landed on heads. What is the probability that the next coin flip will be heads?

• 3/4

• 1/4

• 1/16

• 1/32

• 1/2

Answer explanation

When flipping a fair coin, every flip is an independent event, so the odds of each flip being heads or tails is 1/2. The probability of flipping 5 heads in a row is 1/25 = 1/32— however, in our problem, we have already flipped 4 heads, so the probability of the first four flips being heads is 100%, leaving only the next flip uncertain with a probability of 1/2 for both heads and tails.

What is the probability of being dealt exactly 3 kings in a hand of 5 cards?

• 0.014%

• 0.000017%

• 0.017%

• 0.17%

• 5.77%

Answer explanation

It costs $5 to enter a game where you have a 5% chance of winning $20. What is the expected value of this game?

• −$4.00

• $4.00

• $0.00

• $5.00

• −$5.00

Answer explanation

The expected value of a game is the sum of the probability of every outcome multiplied by that outcome’s value (payout). Since the game costs $5, there is a −$5 cost associated with not winning, which has 95% probability, and there is a ($20 − $5) = $15 net gain associated with winning, which as 5% probability. Therefore the expected value is (0.95)(−$5) + (0.05)($15) = −$4.00.

A $3 lottery either pays nothing or pays $1,000. What is the lowest probability of winning that would still make the lottery worth playing?

• 0.6%

• 0.5%

• 0.03%

• 0.1%

• 0.4%

Answer explanation

Find the variance of a game where you have 25% probability of losing $5, 25% probability of winning $20, and 50% probability of losing $10.

• 220.19

• 5.92

• 53.24

• 160.94

• 64.32

Answer explanation