Back to Exercise: Develop a theoretical probability distribution

Exercises: Develop a Theoretical Probability Distribution and Find Its Expected Value

Work through each section in order. Show your work where indicated. When you build a probability,
separate "the probability of one specific sequence" from "the number of arrangements" that produce
that result. Keep fractions over a common denominator so your sum-to-1 check is easy to read.

Grade 11·24 problems·~45 min·Common Core Math - HS Statistics and Probability·group·hss-md-a-3

Work through problems with immediate feedback

Recall / Warm-Up

On a multiple-choice test each question has four choices and a guesser is correct with
probability $\frac{1}{4}$ . The questions are independent. What is the probability of getting
the first two specific questions both correct?

$\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$

$\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$

$\frac{1}{4}$

$\frac{2}{4} = \frac{1}{2}$

A student guesses on every question of a five-question test where each question has four
equally likely choices. Let $X$ be the number of questions answered correctly. Which statement
correctly describes the model?

$P(\text{correct}) = \frac{1}{4}$ , $P(\text{wrong}) = \frac{3}{4}$ , and the questions are independent.

$P(\text{correct}) = \frac{1}{2}$ because each answer is either right or wrong.

$P(\text{correct}) = \frac{1}{4}$ , but the questions are dependent on one another.

$P(\text{correct}) = \frac{3}{4}$ because there are three wrong choices.

How many different ways are there to choose which 3 of the 5 questions are the correct ones?
That is, evaluate $\binom{5}{3}$ .

Fluency Practice

For the five-question guessing test ( $P(\text{correct}) = \frac{1}{4}$ ), which expression
correctly gives $P(X = 2)$ , the probability of exactly 2 correct?

$\binom{5}{2}\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3$

$\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3$

$\frac{1}{6}$ , since $X$ has six possible values $0,1,2,3,4,5$

$\binom{5}{2}\left(\frac{1}{4}\right)^2$

For the five-question guessing test, compute $P(X = 2) = \binom{5}{2}\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3$ .
Write your answer as a fraction over a denominator of $1024$ .

For the five-question guessing test, compute $P(X = 3) = \binom{5}{3}\left(\frac{1}{4}\right)^3\left(\frac{3}{4}\right)^2$ .
Write your answer as a fraction over a denominator of $1024$ .

The full developed distribution for the five-question guessing test is shown below.

$k$	0	1	2	3	4	5
$P(X=k)$	$\frac{243}{1024}$	$\frac{405}{1024}$	$\frac{270}{1024}$	$\frac{90}{1024}$	$\frac{15}{1024}$	$\frac{1}{1024}$

Add the six numerators to verify the distribution is valid. What is the sum of the numerators?

Using the distribution table above, compute the expected number correct
$E(X) = \sum k \cdot P(X=k)$ for the five-question guessing test. Give a decimal.

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