Peter was given n one-dollar scratch tickets, each winning:

$2 with probability 1/8$10 ------------- 1/100
$1000 ----------- 1/10,000$20000 ---------- 1/1,000,000

1. Let An , where n is a subscript, be the event that Peter wins on at least one of the n tickets he was given. Compute Pr(A8).

2. What is the minimum number of n tickets Peter should be given to ensure that Pr(An)>0.5?

3. Compute the expected value of Peter's winnings when n=8.

How do I do these questions?

2. Originally Posted by maibs89
Peter was given n one-dollar scratch tickets, each winning:

$2 with probability 1/8$10 ------------- 1/100
$1000 ----------- 1/10,000$20000 ---------- 1/1,000,000

1. Let An , where n is a subscript, be the event that Peter wins on at least one of the n tickets he was given. Compute Pr(A8).

2. What is the minimum number of n tickets Peter should be given to ensure that Pr(An)>0.5?

3. Compute the expected value of Peter's winnings when n=8.

How do I do these questions?
The probability that a single ticket wins a prize is $\displaystyle p = \frac{1}{8} + \frac{1}{100} + \frac{1}{10,000} + \frac{1}{1,000,000} = .....$.

Let X be the random variable number of tickets that win a prize.

Then X ~ Binomial(n, p).

1. n = 8: Find Pr(X > 0) = 1 - Pr(X = 0).

2. Require Pr(X > 0) > 0.5 => 1 - Pr(X = 0) > 0.5 => Pr(X = 0) < 0.5.

So solve $\displaystyle {n \choose 0} p^0 (1 - p)^n < 0.5 \Rightarrow (1 - p)^n < 0.5$.

3. Let Y be the random variable amount of money won with 1 ticket.

Then $\displaystyle E(Y) = 2 \left( \frac{1}{8}\right) + 10 \left( \frac{1}{100} \right) + 1,000 \left( \frac{1}{10,000} \right) + 20,000 \left( \frac{1}{1,000,000}\right) = .....$.

The expected amount of money won with 8 tickets is 8 E(Y) = ......