Q: Let $$E_1$$, $$E_2$$, \ldots$$ be an increasing sequence of events. Which of the following is true about their probabilities?

$$\displaystyle \lim_{n \to \infty} P(E_n) = P\left(\bigcup_{i=1}^\infty E_i\right)$$

Q: A deck of 52 cards is dealt into 4 piles of 13 cards each. What is the probability that each pile has exactly one ace?

$$\displaystyle \frac{39 \times 26 \times 13}{51 \times 50 \times 49}$$

Question 1

Suppose a team plays $$n$$ games and wins $$n$$ times and loses $$m$$ times, with all orderings equally likely. What is the probability that there are exactly $$r$$ runs of wins?

Accepted Answer

$$\displaystyle \frac{\binom{m+1}{r} \binom{n-1}{r-1}}{\binom{n+m}{n}}$$

Question 2

Let $$E_1$$, $$E_2$$, \ldots$$ be an increasing sequence of events. Which of the following is true about their probabilities?

Accepted Answer

$$\displaystyle \lim_{n 	o \infty} P(E_n) = P\left(\bigcup_{i=1}^\infty E_iight)$$

Question 3

In the infinite urn paradox, balls numbered $$1$$ through $$10$$ are placed in an urn and a ball is randomly withdrawn at each stage, with $$10$$ new balls added at each step. What is the probability that ball number $$1$$ remains in the urn at $$12 P.M$$.?

Accepted Answer

$$0$$

Question 4

A hospital administrator codes patients as insured $$(1) or $$uninsured $$(0)$$ and as being in good $$(g)$$, fair $$(f)$$, or serious $$(s)$$ condition. What is the size of the sample space for this experiment?

Accepted Answer

$$6$$

Question 5

If $$A$$ and $$B$$ are mutually exclusive events with $$P(A) = 0.3$$ and $$P(B) = 0.5$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.8$$

Question 6

A student is taking a one-hour exam. The probability that the student finishes in less than $$x$$ hours is $$x/2$$ for $$0 \leq x \leq 1. $$Given that the student is still working after $$0.75$$ hours, what is the conditional probability that the full hour is used?

Accepted Answer

$$0.8$$

Question 7

A coin is flipped twice. What is the conditional probability that both flips land on heads, given that at least one flip lands on heads?

Accepted Answer

$$\frac{1}{3}$$

Question 8

In a multiple-choice test with $$m$$ alternatives, a student knows the answer with probability $$p$$ and guesses otherwise. If a student answers correctly, what is the probability they knew the answer?

Accepted Answer

$$\displaystyle \frac{mp}{1 + (m-1)p}$$

Question 9

A blood test detects a disease with $$95\%$$ accuracy when the disease is present, and gives a false positive $$1\% of $$the time. If $$0.5\% of $$the population has the disease, what is the probability that a person who tests positive actually has the disease?

Accepted Answer

$$\displaystyle \frac{95}{294}$$

Question 10

Suppose $$E$$ and $$F$$ are independent events. Which of the following is always true?

Accepted Answer

$$P(E \cap F) = P(E)P(F)$$

Question 11

A parallel system has $$n$$ independent components, with component $$i$$ functioning with probability $$p_i. $$What is the probability the system functions?

Accepted Answer

$$1 - \prod_{i=1}^n (1 - p_i)$$

Question 12

In the gambler's ruin problem, two players $$A$$ and $$B$$ play until one is ruined. If $$A$$ starts with $$i$$ units, $$B$$ with $$N-i$$, and the probability of $$A$$ winning a round is $$p$$, what is the probability $$A$$ wins all the money (for $$p 
eq 1/2$$)?

Accepted Answer

$$\displaystyle \frac{1 - (\frac{1-p}{p})^i}{1 - (\frac{1-p}{p})^N}$$

Question 13

A deck of 52 cards is shuffled. What is the probability that the card following the first ace is a specified card $$x$$?

Accepted Answer

$$\frac{1}{52}$$

Question 14

Suppose $$E$$ and $$F$$ are events with $$P(F) \u003e 0. $$Which formula gives the conditional probability $$P(E|F)$$?

Accepted Answer

$$\displaystyle \frac{P(E \cap F)}{P(F)}$$

Question 15

A bin contains 3 types of flashlights: 20% are type 1 (70% last over 100 hours), 30% are type 2 (40% last over 100 hours), and 50% are type 3 (30% last over 100 hours). If a flashlight lasts over 100 hours, what is the probability it was type 2?

Accepted Answer

$$\frac{12}{41}$$

Question 16

If $$E$$ and $$F$$ are independent, which of the following is also true?

Accepted Answer

$$E is $$independent of $$F^c$$

Question 17

A deck of 52 cards is dealt into 4 piles of 13 cards each. What is the probability that each pile has exactly one ace?

Accepted Answer

$$\displaystyle \frac{39 	imes 26 	imes 13}{51 	imes 50 	imes 49}$$

Question 18

Suppose $$E$$ and $$F$$ are events with $$P(E) = 0.9$$ and $$P(F) = 0.8. $$What is the minimum possible value of $$P(E \cap F)$$?

Accepted Answer

$$0.7$$

Question 19

A box contains $$k+1$$ coins, where the $$ith $$coin lands heads with probability $$i/k. A $$coin is chosen at random and flipped $$n$$ times, all heads. What is the probability the $$(n+1)st $$flip is also heads (for large $$k$$)?

Accepted Answer

$$\displaystyle \frac{n+1}{n+2}$$

Question 20

In the 'three cards' problem, a card is drawn at random from three cards: one red/red, one black/black, one red/black. If the upper side is red, what is the probability the other side is black?

Accepted Answer

$$\frac{1}{3}$$

Conditional Probability, Bayes’s Formula, and Independence: Structured Study Notes

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