Q. 2.37 - A First Course in Probability

Question

An instructor gives her class a set of $10$problems with the information that the final exam will consist of a random selection of $5$them. If a student has figured out how to do $7$the problems, what is the probability that he or she will answer correctly

$\left(a\right)$all $5$problems?

$\left(b\right)$ at least$4$ of the problems?

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right)0.083 \\ \left(b\right)0.5 \end{gathered}$$

Step 1 Given Information.

An instructor gives her class a set of $10$problems with the information that the final exam will consist of a random selection of $5$them.

Syep 2 Part (a) Explanation.

$$\begin{gathered} P(all 5 correctly )=\left(7\mathrm{C}5\right)/\left(10\mathrm{C}5\right) \\ =21 / 252 \\ =0.083 \end{gathered}$$

Step 3 Part (b) Explanation.

$$\begin{gathered} P(at least 4 correctly) =P(exactly 4 correctly)+P(all 5 correctly) \\ =\left[\left(7\mathrm{C}4{)}^{*}\right(3\mathrm{Cl})\right]/\left(10\mathrm{C}5\right)+\left(7\mathrm{C}5\right)/\left(10\mathrm{C}5\right) \\ =\left[\left({35}^{*}3\right)/252\right]+[21/252] \\ =[105 / 252]+[21 / 252] \\ =126 / 252 \\ =0.5 \end{gathered}$$