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Quantitative Methods
MA 103
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Probability Worksheet
Name:
KEY
1. One card is drawn at random from a well-shuffled standard deck of 52 cards.
Let H represent the event “the card is a Heart.”
Let C represent the event “the card is a Club.”
Let K represent the event “the card is a King.”
Let Q represent the event “the card is a Queen.”
Let A represent the event “the card is an Ace.”
Let T represent the event “the card is a two.”
(a) Find Pr(H).
13/52 = 0.25
(b) Find Pr(A).
4/52 = 0.0769
(c) Find Pr(K).
4/52 = 0.0769
(d) Explain why Pr(H or A) does NOT equal Pr(H) + Pr(A), and find Pr(H or A).
The events H and A are not mutually exclusive (or disjoint), since we have the “Ace of Hearts,” so they
overlap.
Pr(H or A) = Pr(H) + Pr(A) – Pr(H and A)
Pr(H or A) = 0.25 + 0.0769 – 1/52
Pr(H or A) = 0.25 + 0.0769 – 0.019 = 0.30769
(e) Explain why Pr(A or T) does equal Pr(A) + Pr(T).
The events A and T are mutually exclusive (disjoint), since a card can not be an Ace and a two.
(f) Find Pr(H and C)
=0 since disjoint
(g) Find Pr(H or C).
In general:
Pr(A or B) = Pr(A) + Pr(B) – Pr(A and B)
Pr(H or C) = Pr(H) + Pr(C) – Pr(H and C)
Pr(H or C) = 0.25 + 0.25 – 0 = 0.50
2. A senior at QM College begins looking for jobs, and reads that for his POE, starting salaries are normally
distributed, with a mean salary of $43,300, and a standard deviation of $2,700.
(a) How likely is he to find a job paying more than $46,000?
z = (46000-43300)/2700 = 1.0. The empirical rule says that 68% of the salaries are +/- 1 std. dev. from the
mean, so 34% of the salaries would be +1 std. dev. The 50% to the left of the mean, plus this addl. 34%, leaves
100 – (50+68/2) = 16% of the salaries.
(b) How likely is he to find a job paying more than $37,900?
z = (37900-43300)/2700 = -2.0. The empirical rule says that 95% of the salaries are +/- 2 std. dev. from the
mean, so that leaves 100 – 95% = 5% of the salaries to outside this domain. Since normal distributions are
symmetric, this 5% is split, leaving 2.5% greater than mean +2 std. dev. and 2.5% less than mean-2 std. dev.
So, 97.5% of all salaries will be greater than $37,900.
3. The following is a table of votes cast in the U.S. Senate:
Let Y represent the event “the Senator voted Yea.”
Democrat
Republican
Total
Let N represent the event “the Senator voted Nay.”
45
15
60
Yea
Let R represent the event “the Senator is a Republican.”
5
35
40
Nay
Let D represent the event “the Senator is a Democrat.”
50
50
100
Total
(a) Find Pr(Y).
60/100 = 0.60
(b) Find Pr(Y and R).
15/100 = 0.15
(c) Find Pr(Y or R).
Pr(Y or R) = Pr(Y) + Pr(R) – Pr(Y and R)
Pr(Y or R) =0.60+0.50 – 0.15 = 0.95
Note that there are 5 Dem’s who voted nay…
(c) Find Pr(Y given R).
15/50 = 0.30
(d) Are the events Y and R independent?
No, Pr(Y given R)=0.30, and Pr(Y)=0.60, different probabilities, so Y and R are NOT independent.