HW # 1.10 Probability Distribution
1. A fair coin is thrown four times. The random variable X is the number of heads obtained.
a) Tabulate the probability distribution of X.
b) P(X = 0) =
c) P(X > 2) =
d) P(X > 2│X > 1)
2. Tim picks a card at random from an ordinary pack. If the card is an ace, he stops; if not, he continues to
pick cards at random, without replacement, until either an ace is picked, or four cards have been
drawn. The random variable C is the total number of cards drawn. Construct a tree diagram to
illustrate the possible outcomes of the experiment, and use it to calculate the probability distribution
of C.
3. A bag contains six red and three green counters. Two counters are drawn from the bag without
replacement. Tabulate the probability distribution of the number of green counters obtained.
4. When a four-sided dice is thrown, the score is the number on the bottom face. Two fair four-sided
dice, each with faces numbered 1 to 4, are thrown simultaneously. The random variable M is the
product of the two scores multiplied together. Tabulate the probability distribution of M, combining
together all the possible ways of obtaining each value.
5. Calculate the value of α for the following probability distribution table.
x
P(X = x)
6.
y
P(Y = y)
0
α
1
β
1
2α
2
2β
2
3α
3
3β
4
4β
a) Find the value of β.
b) P(Y = 2) =
c) P(Y > 2) =
7. Review: Events A and B have probabilities P(A) = 0.4, P(B) = 0.65, and P(A U B) = 0.85.
a) Calculate P(A  B)
b) Calculate P(B │ A)
c) Are events A and B independent? Justify your reasoning.
d) Are events A and B mutually exclusive? Explain.
HW #1.11 Expected Value
1. Find the mean of the random variables X and Y which have the following probability distributions.
a)
x
P(X = x)
b)
y
P(Y = y)
0
1
8
1
3
8
2
1
8
3
1
4
4
1
8
-2
0.15
-1
0.25
0
0.3
1
0.05
2
0.2
3
0.05
2. Find the exact expectation of the random variable Y, which has the following probability distribution.
y
3
4
5
6
7
1
5
7
1
4
P(Y = y)
18
18
18
18
18
3. The six faces of a fair cubical dice are numbered 1, 2, 2, 3, 3, and 3. When the dice is thrown once, the
score is the number appearing on the top face. This is denoted by X.
a) Find the mean of X.
b) The dice is thrown twice and Y denotes the sum of the scores obtained. Tabulate the probability
distribution of Y. Hence, find E(Y).
4. Bag A contains 2 red balls and 3 green balls. Two balls are chosen at random from the bag without
replacement. Let X denote the number of red balls chosen. The following table shows the probability
distribution for X.
x
0
1
2
3
1
P(X = x)
a
10
10
a) Find the value of a.
b) Calculate E(x), the mean number of red balls chosen.
Bag B contains 4 red balls and 2 green balls. Two balls are chosen at random from bag B.
c) Draw a tree diagram to represent the above information, including the probability of each
event.
d) Hence, find the probability distribution for Y, where Y is the number of red balls chosen.
A standard die with six faces is rolled. If a 1 or a 6 is obtained, two balls are chosen from bag A.
Otherwise two balls are chosen from bag B.
e) Calculate the probability that two red balls are chosen.
f) Given that two red balls are obtained, find the conditional probability that a 1 or a 6 was rolled
on the die.