If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is $i$ for $i=1,\dots,12$?
Consider an urn containing 12 balls, of which 8 are white. A sample of size 4 is to be drawn with replacement (without replacement). What is the conditional probability (in each case) that the first and third balls drawn will be white given that the sample drawn contains exactly 3 white balls?
Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the conditional probability that the first card selected is a spade given that the second and third cards are spades.
In a certain community, 36% of the families own a dog and
22% of the families that own a dog also own a cat. In addition, 30% of
the families own a cat. What is
a. the probability that a randomly selected family owns both a
dog and a cat?
b. the conditional probability that a randomly selected family
owns a dog given that it owns a cat?
Urn A contains 2 white and 4 red balls, whereas urn B contains 1 white
and 1 red ball. A ball is randomly chosen from urn A and put into urn
B, and a ball is then randomly selected from urn B. What is
a. the probability that the ball selected from urn B is white?
b. the conditional probability that the transferred ball was
white given that a white ball is selected from urn B?
a. A gambler has a fair coin and a two-headed coin in his pocket. He
selects one of the coins at random; when he flips it, it shows
heads. What is the probability that it is the fair coin?
b. Suppose that he flips the same coin a second time and, again, it
shows heads. Now what is the probability that it is the fair coin?
c. Suppose that he flips the same coin a third time and it shows
tails. Now what is the probability that it is the fair coin?
Let $E\subset F$. Express the following probabilities as simply as possible $$ P(E|F),\: P(E|F^\mathsf{c}),\: P(F|E),\: P(F|E^\mathsf{c}). $$
A ball is in anyone of $n$ boxes and is in the $i$-th box with probability $P_i$. If the ball is in box $i$, a search of that box will uncover it with probability $ \alpha _i$. Show that the conditional probability that the ball is in box $j$, given that a search of box $i$ did not uncover it, is $$ \frac{P_j}{1-\alpha _iP_i},\text{ if }j\neq i,\text{ and } \frac{(1-\alpha _i)P_i}{1-\alpha_iP_i},\text{ if }i=j. $$
An event is said to carry negative information about an event $E$ iff
$$
P(E|F)\leq P(E).
$$
This is denoted by $F\searrow E$. Either give a proof or provide a
counterexample for the following assertions
a. If $F\searrow E$, then $E\searrow F$.
b. If $F\searrow E$ and $E\searrow G$, then $F\searrow G$.
c. If $F\searrow E$ and $G\searrow E$, then $F\cap G\searrow G$.
The probability of getting a head on a single toss of a coin is $p$. Suppose that A starts and continues to flip the coin until a tail shows up, at which point B starts flipping. Then B continues to flip until a tail comes up, at which point A takes over, and so on. Let $P_{n,m}$ denote the probability that A accumulates a total of $n$ heads before B accumulates $m$. Show that $$ P_{n,m}=p\, P_{n-1,m}+(1-p)(1-P_{m,n}) . $$