## Math 3118 --- Ch 7 Problems

Wednesday 1/19
7.1: 3, 6,7,10 (do straight flush, flush and full house), 12

1. Powerball. Balls numbered 1-49 are placed in a container and 5 of them are drawn out of the container without replacement. Then a separate "power ball" is drawn from another contained containing balls numbered 1-42. Meanwhile, you write down 5 numbers from 1-49 and one number from 1-42 on a card. To win the jackpot, your first 5 numbers must match the numbers of the first 5 balls (not necessarily in order) and your 6th number must match the number of the 6th ball (the "powerball"). What is the probability that you will win ?

2. You can view a birthday as a random choice of a number from 1-365. Let E be the event that in a room of 23 people, at least two have the same birthday.
a. Describe the complementary event Ec.
b. Compute the probability P(Ec) and use it to find P(E).

HW 7.1: 8,9,10 (do four of a kind)

Monday 1/24
7.1: 15, 16
7.2: 3 (see below), 4ab, 5, 8 (see below)
7.3: 1 ii,iv,vi

P(1)=1/21 P(2)=2/21 P(3)=3/21 P(4)=4/21 P(5)=5/21 P(6)=6/21
We get the 21 from 21 = 1+2+3+4+5+6. For the sample points for a pair of dice, just multiply the probabilites for the individual dice. For example the probability that the first die is a 2 and the second is a 5 would be
P(2,5) = 2/21 x 5/21 = 10/441

More about 7.2.8: let E and F be the events
E = "first die is a 3" F = "second die is a 3"
Describe the events E»F and E«F both in words and by listing the elements of each event. Then verify formula 7.1 by separately computing all four of the probabilities in it.

HW: 7.2.4cd and 7.3.1 iii

Wednesday 1/26
7.3: 4, 5, 6, 9, 11
7.6: 3, 4

HW 7.3: 10 , 7.6: 5

Monday 1/31
7.6: 6, 7i,ii, 11, 15, 17, 16 (see below)

HW 7.6: 7 iii,iv + Study for test

Here is one stategy for solving problem 7.6.16, the notorious "Monty Hall Problem"

You, the contestant, choose door number 2. Now identify several events.
C1 = Car behind door number 1
C2 = Car behind door number 2
C3 = Car behind door number 3
It is reasonable to assign probability 1/3 to each of these.
M1 = Monty opens door number 1
M3 = Monty opens door number 2
Of course Monty won't open door number 2. Also (and this is the source of the difficulty) Monty will not open the door with the car behind it (and he knows which door that is). The information about Monty's behavior can be summarized using conditional probabilities as follows:
P(M1 | C3) = 1 P(M1 | C1) = 0
P(M3 | C1) = 1 P(M3 | C3) = 0
In other words given that the car is behind C3 Monty will definitely open door number 1 and vice versa. If the car is really behind door 2 Monty chooses which of doors 1 or 3 to open at random so
P(M1 | C2) = 1/2 P(M3 | C2) = 1/2
Now suppose Monty opens door number 1. Should you stick with door 2 or switch to door 3 or doesn't it matter? To find out, compute the conditional probabilities:
P(C2 | M1) and P(C3 | M1)
to see which is larger.

Wednesday 2/2 ---- Chapter 7 Test