# High School: Statistics and Probability

### Using Probability to Make Decisions HSS-MD.A.3

3. Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

Students should be able to find expected values and distinguish random variables already. Now, they should do all that while standing on their heads and juggling coconuts.

Well, maybe not. But this standard does ask students to do a little bit more than know what values are and how to find them. Students should think through the problem and develop their own theoretical probabilities (instead of being spoon-fed the theoretical probabilities of a question).

This means that students should be able to distinguish when events have equal probabilities or differing ones, as well as which are associated with which. They should be able to figure out what they're being asked to find (the expected value or the probability or the actual average) and calculate it accordingly.

Yes, this involves using both P(X = a) = nCapaqna and E(x) = x1p1 + x2p2 + … + xipi. Tell your students not to get too excited.

Once they have achieved the ability to understand theoretical probabilities, they must then apply the lessons of the previous two standards to come up with expected value and probability distributions. The next step after that is coconuts and an impeccable sense of balance (and a comprehensive health insurance plan).

#### Drills

1. One of the dangers of driving on the road, especially at high speed, is getting a flat tire. Assuming the tire blowout is completely random, what is the theoretical probability that it will happen?

1 out of 4

Cars have four tires (most of them do, anyway). Since we assume the blowout of a tire is a random process, each tire has an equal chance of a blowout. That means a 1 in 4 chance. So our answer is (C).

2. Your chances of getting hit by lightning on planet Electrick are 1 out of 15 visits. It's not the ideal vacation getaway. What are the chances you'll be struck two times or fewer in 15 brief visits to the planet?

0.93

Our probability of getting struck by lightning is 1 out of 15, or 0.0667. We can create a probability distribution with n = 15, p = 0.0667, q = 0.9333, and a going from 0 to 2 (because we're interested in being struck by lighting two or fewer times). This will give us P(0) = 0.355264, P(1) = 0.38064, and P(2) = 0.19032. The probability of all of those happening is the sum of the probabilities, or 0.926. The chances that you'll be struck by lightning two or fewer times in 15 visits are about 93%.

3. What is the probability that you will answer this multiple choice question (with 4 answer choices) correctly completely at random and answer the next answer choice incorrectly completely at random (also 4 answer choices)?

0.1875

The probability of answering a question incorrectly, assuming one correct answer, is 0.75. The probability of answering correctly at random is 0.25. If we multiply the probabilities together, that will give us the probability of giving a correct answer and then an incorrect answer. Or an incorrect answer and then a correct answer. Either way, order matters. That means our answer is 0.25 • 0.75 = 0.1875. Since using the P(X = a) equation doesn't care about order, our answer can also be found by dividing P(X = 1) in half.

4. An economist estimates a stock that costs \$10 today to reach \$15. If the stock's price is completely random (meaning the economist knows nothing), the value of the stock can just as easily reach \$0. What is the expected value of investing \$100 dollars?

-\$50

We stand to lose \$100 if the stock goes to \$0 or, if we believe our economist, we could gain \$50. Since the stock's price is random, the probability of each occurring is 0.5. The average of the two would then be 0.5 • -100 + 0.5 • 50 = -25. That negative sign tells you this is a bad investment decision.

5. A random generator is able to pick out 1 letter from the English alphabet at any given time. What is the probability that this generator will pick out the same letter three times in a row at any point in 26 randomly generated letters?

5.69 × 10-5

Since we know the probability of selecting one letter randomly is 1 in 26, doing so 3 times in a row gives a probability of . The key here is that the letters are chosen randomly and in a row, so the formula P(X = 3) doesn't apply; it would give the probability of the same letter chosen at any three points within the 26 trials.

6. The probability that you will have a son is 0.5. (This isn't exactly true in real life, but we'll assume it is.) If you plan on having four children total, what is your expected value of sons?

E(x) = 2

The sex of each child is an individual and random event, so we can multiply each of the four events by the probability of 0.5. That means our expected value is 1 • 0.5 for each child means 4 • 1 • 0.5 = 2. The highest probability would be when X = 2, so this is in line with our probability distribution.

7. Every time an astronaut is out during a spacewalk, she has a 1 in 5 chance of being exposed to damaging radiation. For what a value is P(X = a) the greatest out of five walks?

1

A 1 in 5 chance translates to a p value of 0.2 and a q = 0.8. Our n value is 5, and a goes from 1 to 5. The largest P(X = a) is when a = 1. This makes sense even with the information provided in the problem alone. The probability is 1 out of 5 walks, so it stands to reason that the highest P(X = a) value would occur when a = 1.

8. The probability that a hurricane will destroy a building is 10%. What is our expected value of total buildings destroyed after a hurricane hits a town with 10,000 buildings?

1,000

All we need to do is use the formula E(x) = x1p1 + x2p2 + … + xipi. We have 10,000 buildings, so all we have to do is multiply the number of buildings by the probability of their destruction. E(x) = 10,000 • 0.1 = 1,000 buildings.

9. A probability distribution for someone picking out their favorite color by choice is unlikely to be correct because most likely because of which of the following?

You did not select anything randomly

This question asks you which is most likely the case. The sample space could be whatever colors are available, and there could be probability terms assigned to colors. Consciously choosing a color without randomizing the process would lead to an invalid probability distribution specifically because of the lack of randomization.

10. An expected value is unlikely to be correctly calculated if which of the following occurs?