Honors Mathematics I

Let S be the sample space {1, 2, 3, 4, 5}. Let E be the event {1, 2}, F be the event {2, 4, 6} and G be the event {2, 3, 5}. Find the events:

1. E F
2. E F
3. G^c F

Let S, E, F and G be as in the previous question, and let the probability of the sample points be given by the following table:

1. Find P(E), P(F) and P(G)
2. Find P(E F)
3. Find P(E F) and verify the inclusion-exclusion rule for porbability.

A coin is tossed 5 times. What is the probability of:

1. Tossing H, H, T, H, H in that order?
2. Tossing 4 heads and one tail in any order?
3. Tossing at least one tail?
4. Tossing at least four heads?

What is the probability that in a group of 10 friends, at least 2 share a birthday? (Ignore leap years).

In poker, a straight flush is a sequence of 5 cards of the same suit in order (eg. 10S, JS, QS, KS, AS), while a straight is a sequence of 5 cards in order, but not all of the same suit (eg. AS, 2S, 3D, 4H, 4S). Aces can be either high or low.

1. What is the probability that a player is dealt a straight flush?
2. What is the probability that a player is dealt a flush (but not a straight flush)?

In a state lottery, players guess which 6 numbers from 1 to 36 will be drawn at random. Order does not matter, and numbers cannot be picked twice.

• What is the probability that a player gets all 6 numbers correct?
• What is the probability that a player gets exactly 5 numbers right?
• What is the probability that a player gets no numbers right?
• What is the probability that a player gets at least one number right?

Two cards are drawn at random from a deck. If the first is an ace, what is the probability that the second card is an ace? Verify that the product rule for probability works in this case, by comparing this probability with the probability that the first card is an ace and the probability of drawing two aces.

What is the probability that the second card is a 10, J, Q or K?

In a family with three children, what is the probability that all three children are girls, given that it is known that one child is a girl? (Assume a child is equally likely to be a boy or a girl.)

A web hosting company estimates that a typical web server will be unavailable (because of a crash or maintenance) an average of 1% of the time. Assuming that whether or not two different servers are unavailable are independent events, how many servers are required to ensure that the probability of at least one of the servers being available at any time is at least 99.999%?

A red and a green dice are rolled. Show that the event where the roll of the red dice is 1 is independent from the event where the total on the two dice is 7.

Under what conditions are mutually exclusive events independent?