A worker has asked her supervisor for a letter of recommendation for a
new job. She estimates that there is an 80% chance that she
will get the job if she receives a strong recommendation, a 40%
chance if she receives a moderately good recommendation, and a 10%
chance if she receives a weak recommendation. She further
estimates that the probabilities that the recommendation will be
strong, moderate, and weak are 0.7, 0.2, and 0.1,
respectively.
a. How certain is she that she will receive the new job offer?
b. Given that she does receive the offer, how likely should she
feel that she received a strong recommendation? A moderate
recommendation? A weak recommendation?
c. Given that she does not receive the job offer, how likely
should she feel that she received a strong recommendation? A moderate
recommendation? A weak recommendation?
Barbara and Dianne go target shooting. Suppose that each of Barbara's
shots hits a wooden duck target with probability $p_1$, while each shot
of Dianne's hits it with probability $p_2$. Suppose that they shoot
simultaneously at the same target. If the wooden duck is knocked over
(indicating that it was hit), what is the probability that
a. both shots hit the duck?
b. Barbara's shot hit the duck?
In successive rolls of a pair of fair dice, what is the probability of getting 2 sums of seven before 6 even numbers?
An urn contains 12 balls, of which 4 are white. Three players (A, B,
and C) successively draw from the urn in that order repeatedly. The
winner is the first one to draw a white ball. Find the probability of
winning for each player if
a. each ball is replaced after it is drawn.
b. the balls that are withdrawn are not replaced.
Independent trials that result in a success with probability $p$ are successively performed until a total of $r$ successes is obtained. Show that the probability that exactly $n$ trials are required is $$ {n-1\choose r-1}p^r(1-p)^{n-r} $$
Independent trials that result in a success with probability $p$ and a failure with probability $1-p$ are called Bernoulli trials. Let $P_n$ denote the probability that $n$ Bernoulli trials result in an even number of successes (0 being considered an even number). Show that $$ P_n=p(1-P_{n-1})+(1-p)P_{n-1}\text{ for }n\geq 1, $$ and use this formula to prove (by induction) that $P_n=\frac{1+(1-2p)^n}{2}$.
Let $Q_n$ denote the probability that no run of 3 consecutive heads appears in $n$ tosses of a fair coin. Show that $$ Q_n = \frac{1}{2}Q_{n-1}+\frac{1}{4}Q_{n-2}+\frac{1}{8}Q_{n-3},\: Q_0=Q_1=Q_2=1. $$ Find $Q_8$.
Give a direct proof of the fact that $$ P(E|F)=P(E|F\cap G)P(G|F)+P(E|F\cap G^\mathsf{c})P(G^\mathsf{c}|F). $$ for events $E,F,G$ of a probability space.
Extend the definition of conditional independence to more than 2 events.
Show that $$ P(E|E\cup F)\geq P(E|F) $$ for any events $E$ and $F$ of a probability space.