Probability — Practice

Question 1

Two dice are thrown. Find the probability that the sum is at least $10$.

Accepted Answer

Sums of $10$, $11$ and $12$ arise from $3$, $2$ and $1$ ordered pairs respectively.

$\dfrac{6}{36}=\dfrac16$.

Question 2

From a pack of $52$ cards, $5$ are drawn. Find the probability that exactly two are aces.

Accepted Answer

$\dfrac{\binom42\binom{48}{3}}{\binom{52}{5}}=\dfrac{6	imes17296}{2598960}$.

$=\dfrac{103776}{2598960}\approx0.0399$.

Question 3

$P(A)=0.3$, $P(B)=0.4$ and $A$, $B$ are independent. Find $P\left(A\cup B\right)$ and $P\left(A'\cap B'\right)$.

Accepted Answer

$P\left(A\cap Bight)=0.12$, so $P\left(A\cup Bight)=0.3+0.4-0.12=0.58$.

$P\left(A'\cap B'ight)=0.7	imes0.6=0.42$, which is $1-0.58$ as it must be.

Question 4

Three friends each solve a problem independently with probabilities $	frac12$, $	frac13$ and $	frac14$. Find the probability that the problem is solved.

Accepted Answer

$P\left(	ext{none solves}ight)=\dfrac12\cdot\dfrac23\cdot\dfrac34=\dfrac14$.

$P\left(	ext{solved}ight)=1-\dfrac14=\dfrac34$.

Question 5

A card is drawn from a pack and found to be a face card. Find the probability that it is a king.

Accepted Answer

There are $12$ face cards, of which $4$ are kings.

$P\left(	ext{king}\mid	ext{face}ight)=\dfrac{4}{12}=\dfrac13$.

Question 6

Urn A has $2$ white and $3$ black balls; urn B has $4$ white and $1$ black. An urn is chosen at random and a white ball drawn. Find the probability that urn A was chosen.

Accepted Answer

Path A: $\dfrac12\cdot\dfrac25=\dfrac15$. Path B: $\dfrac12\cdot\dfrac45=\dfrac25$.

$P\left(A\mid	ext{white}ight)=\dfrac{1/5}{1/5+2/5}=\dfrac13$.

Urn B has twice the white proportion, so a white draw makes it twice as likely.

Question 7

In a town, $60\%$ of people read newspaper X and $40\%$ read Y, with $20\%$ reading both. A randomly chosen reader of X is found. Find the probability that they also read Y.

Accepted Answer

$P\left(Y\mid Xight)=\dfrac{P\left(X\cap Yight)}{P(X)}=\dfrac{0.2}{0.6}=\dfrac13$.

Since $P(X)P(Y)=0.24
eq0.2$, the two are not independent.

Question 8

A test detects a condition with probability $0.95$ when present and gives a false positive with probability $0.02$. The condition affects $1\%$ of people. Find the probability that a positive result is genuine.

Accepted Answer

True positives: $0.01	imes0.95=0.0095$.

False positives: $0.99	imes0.02=0.0198$.

$P\left(	ext{condition}\mid+ight)=\dfrac{0.0095}{0.0095+0.0198}\approx0.324$.

Even a good test misfires more often than it hits when the condition is rare.

Question 9

Show that if $A$ and $B$ are independent with $P(A)=0.5$ and $P(B)=0.6$, they cannot be mutually exclusive.

Accepted Answer

Independence gives $P\left(A\cap Bight)=0.5	imes0.6=0.3$.

Mutual exclusivity would require $P\left(A\cap Bight)=0$.

The two are incompatible whenever both probabilities are positive.

Question 10

A random variable takes values $0,1,2,3$ with probabilities $0.1$, $0.3$, $0.4$, $0.2$. Find its mean and variance.

Accepted Answer

$E(X)=0+0.3+0.8+0.6=1.7$.

$E\left(X^{2}ight)=0+0.3+1.6+1.8=3.7$.

$\operatorname{Var}(X)=3.7-2.89=0.81$.

Question 11

A fair coin is tossed $8$ times. Find the probability of exactly $3$ heads, and of at least $1$ head.

Accepted Answer

$P(X=3)=\binom83\left(	frac12ight)^{8}=\dfrac{56}{256}=\dfrac{7}{32}$.

$P\left(X\ge1ight)=1-\left(	frac12ight)^{8}=\dfrac{255}{256}$.

Question 12

A binomial variable has mean $6$ and variance $2$. Find $P(X=0)$.

Accepted Answer

$np=6$ and $npq=2$ give $q=	frac13$, so $p=	frac23$ and $n=9$.

$P(X=0)=q^{9}=\left(	frac13ight)^{9}=\dfrac{1}{19683}$.

A check: the variance $2$ is less than the mean $6$, as it must be.

Practice — Probability

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