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Exercises Probability and the bell curve

Example 3.1 from Statistics, 3.1 Terminology

Terminology: example 3.1

The sample space S is the whole numbers starting at one and less than 20.

S = ________
Let event A = the even numbers and event B = numbers greater than 13.
A = ________, B = ________
P(A) = ________, P(B) = ________
A AND B = ________, A OR B = ________
P(A AND B) = ________, P(A OR B) = ________
A′ = ________, P(A′) = ________
P(A) + P(A′) = ________
P(A|B) = ________, P(B|A) = ________; are the probabilities equal?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}
2. A = {2, 4, 6, 8, 10, 12, 14, 16, 18}, B = {14, 15, 16, 17, 18, 19}
3. P(A) = $\frac{number of outcomes in A}{number of outcomes in S}$ = $\frac{9}{19}$ , P(B) = $\frac{number of outcomes in B}{number of outcomes in S}$ = $\frac{6}{19}$
4. The set A AND B contains all outcomes that lie in both sets A and B, so A AND B = {14,16,18}, The set A OR B contains all outcomes that lie either of the sets A or B, so A OR B = {2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19}.
5. P(A AND B) = $\frac{3}{19},$ P(A OR B) = $\frac{12}{19}$
6. A' consists of all outcomes in the sample space, S, that DO NOT lie in A, so A′ = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19; P(A′) = $\frac{10}{19}$ .
7. P(A) + P(A′) = $\frac{9}{19}$ + $= \frac{\frac{3}{19}}{\frac{9}{19}}$ = 1
8. P(A|B) = $\frac{P (A AND B)}{P (B)}$ $= \frac{\frac{3}{19}}{\frac{6}{19}}$ = $\frac{3}{6}$ , P(B|A) = $\frac{P (A AND B)}{P (A)}$ $= \frac{\frac{3}{19}}{\frac{9}{19}}$ = $\frac{3}{9}$ , No, the probabilities are not equal.

How did it go?

Next Terminology: example 3.2

From Statistics (High school edition) by OpenStax (Rice University), section 3.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.