Probability: Mixed Practice

Before you start#

Here are twelve ques­tions from across the whole chap­ter. Before you write any­thing, pause and ask your­self what kind of ques­tion it is. Is it a con­di­tional prob­a­bil­ity? A prod­uct of inde­pen­dent events? Or do you need Bayes' the­o­rem to turn a con­di­tion around? Get that choice right and the work­ing usu­ally falls into place.

Try these#

1. $P(A) = 0.7$, $P(B) = 0.5$, $P(A \cup B) = 0.85$. Find $P(A \cap B)$, $P(A \mid B)$ and then decide whether $A$ and $B$ are inde­pen­dent.
2. Two cards are drawn with­out replace­ment. Find the prob­a­bil­ity that both are aces, and the prob­a­bil­ity that the sec­ond is an ace given that the first was not.
3. You keep throw­ing a die until a six turns up. Find the prob­a­bil­ity that it takes exactly three throws.
4. Three friends try a puz­zle on their own, with suc­cess prob­a­bil­i­ties $\displaystyle \tfrac{1}{2}$, $\displaystyle \tfrac{1}{3}$ and $\displaystyle \tfrac{1}{5}$. Find the prob­a­bil­ity that exactly one solves it.
5. A fam­ily has three chil­dren. Given that at least one is a boy, find the prob­a­bil­ity that all three are boys.
6. In a class, $45\%$ of the stu­dents are girls; $20\%$ of girls and $30\%$ of boys wear glasses. A stu­dent with glasses is cho­sen at ran­dom. Find $P(\text{girl})$.
7. Bag A has $4$ red and $6$ black balls; bag B has $7$ red and $3$ black. A die is thrown: on $1$ or $2$ a ball is drawn from A, oth­er­wise from B. The ball is red. Find $P(\text{from A})$.
8. $A$ and $B$ are inde­pen­dent, $P(A) = 0.4$ and $P(A \cap B) = 0.16$. Find $P(B)$ and $P(A' \cap B')$.
9. An insur­ance com­pany insures $3000$ scooter rid­ers, $5000$ car dri­vers and $2000$ truck dri­vers, with acci­dent prob­a­bil­i­ties $0.02$, $0.04$ and $0.10$. One of these insured peo­ple has an acci­dent. Find the prob­a­bil­ity that it was a scooter rider.
10. A coin is tossed. If it shows heads, a die is thrown and the num­ber noted. If it shows tails, a card is drawn from a pack. Find the prob­a­bil­ity of a head fol­lowed by a six.
11. Prove that mutu­ally exclu­sive events $A$, $B$ with $P(A), P(B) > 0$ can­not be inde­pen­dent.
12. A shelf has $3$ faulty and $7$ good bat­ter­ies. Two are tested one after the other, with­out replace­ment. If the sec­ond one is good, find the prob­a­bil­ity that the first was faulty.

Answers to check your work#