Back to Exercise: Analyze decisions and strategies using probability

Exercises: Analyze Decisions and Strategies Using Probability

Work through each section in order. For testing problems, build the two-way
table over the stated population and compute the decision-relevant quantity by
counting. Remember the difference between a test's accuracy P(positive | disease)
and the decision-relevant P(disease | positive). For strategy problems, account
for how a choice changes ALL of the probabilities, not just the one you want.
Write explanation answers in complete sentences and name your assumptions.

Grade 12·20 problems·~35 min·Common Core Math - HS Statistics and Probability·group·hss-md-b-7

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Warm-Up: Tools You Already Have

These problems review expected value and conditional probability, the tools this lesson combines.

A game has expected value $\text{\char"0024}2$ per play. Which statement best describes what the expected value tells you?

Over many plays, your average gain per play tends toward $\text{\char"0024}2$ ; it does not guarantee the result of any single play.

You will gain exactly $\text{\char"0024}2$ every time you play.

Your largest possible single-play gain is $\text{\char"0024}2$ .

The probability of winning is $2$ .

A coach is deciding whether to attempt a risky play or a safe play at the end of a game. Which list correctly names the four ingredients of a decision analysis?

The competing strategies, the uncertain outcomes, the probabilities of those outcomes, and the payoffs.

Only the expected value of each choice.

The final score and nothing else.

The coachs reputation and the fans opinions.

In a population of $1000$ people, a two-way table shows $40$ people have a condition and test positive, while $60$ people do NOT have the condition but also test positive. Among all who test positive, how many actually have the condition? ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ And how many test positive in total? ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

Fluency Practice

Compute the decision-relevant quantity in each problem.

Two insurance choices have the SAME expected annual cost of $\text{\char"0024}500$ . Plan X costs exactly $\text{\char"0024}500$ every year. Plan Y costs $\text{\char"0024}0$ with probability $0.98$ but $\text{\char"0024}25{,}000$ with probability $0.02$ . What is the largest possible single-year cost (the worst case) under Plan Y, in dollars?

A disease has prevalence $1\%$ in a population of $10{,}000$ people, so $100$ have it and $9{,}900$ do not. A test is $99\%$ sensitive (catches $99$ of the $100$ sick) and $99\%$ specific (correctly clears $99\%$ of the healthy). How many of the $9{,}900$ healthy people test positive (false positives)?

Using the table from the previous problem: $99$ true positives and $99$ false positives test positive, for $198$ positives in all. What is $P(\text{disease} \mid \text{positive})$ , written as a decimal?

A team trailing by one goal late can KEEP its goalie or PULL the goalie. Which statement correctly describes the effect of pulling the goalie?

It raises the team’s chance of scoring the tying goal AND raises the opponent’s chance of an empty-net goal.

It only raises the team’s chance of scoring; it has no effect on the opponent.

It only helps the opponent and never helps the trailing team.

It changes nothing about either team’s probabilities.

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