Question

probability rules practice worksheet - fall 2025 - statistics - room 404
for each problem listed below, find p(a), p(b), p(a), p(b), p(a∩b), p(a∪b), and m.e.?
a number is randomly chosen from 1 to 20.

1. a: # greater than 15 b: multiple of 4
2. a: even # b: 11

a single card is randomly chosen from a deck.

1. a: black b: jack
2. a: face b: club
3. a: ace b: number

two dice are rolled.

1. a: sum of 8 b: at least one of the dice is a 5
2. a: sum of 5 or lower is rolled b: the first die rolled is a 6
3. given the following venn diagram and associated probabilities of sample points. p(1)=.28 p(2)=.12 p(3)=.18 p(4)=.42
4. given the following chart(a: female, b: vote no)

female male
yes 24 9 33
no 9 18 27
33 27 60

Explanation:

1.

Step1: Calculate P(A)

There are 5 numbers greater than 15 in the set {1, 2, ..., 20}, so $P(A)=\frac{5}{20}=\frac{1}{4}$.

Step2: Calculate P(B)

There are 5 multiples of 4 in the set {1, 2, ..., 20} (4, 8, 12, 16, 20), so $P(B)=\frac{5}{20}=\frac{1}{4}$.

Step3: Calculate P(A')

$P(A') = 1 - P(A)=1-\frac{1}{4}=\frac{3}{4}$.

Step4: Calculate P(B')

$P(B') = 1 - P(B)=1 - \frac{1}{4}=\frac{3}{4}$.

Step5: Calculate P(A∩B)

The numbers that are greater than 15 and multiples of 4 are 16 and 20, so $P(A\cap B)=\frac{2}{20}=\frac{1}{10}$.

Step6: Calculate P(A∪B)

Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, we have $P(A\cup B)=\frac{1}{4}+\frac{1}{4}-\frac{1}{10}=\frac{5 + 5- 2}{20}=\frac{8}{20}=\frac{2}{5}$.

Step1: Calculate P(A)

There are 10 even - numbered elements in the set {1, 2, ..., 20}, so $P(A)=\frac{10}{20}=\frac{1}{2}$.

Step2: Calculate P(B)

There is 1 element equal to 11 in the set {1, 2, ..., 20}, so $P(B)=\frac{1}{20}$.

Step3: Calculate P(A')

$P(A') = 1 - P(A)=1-\frac{1}{2}=\frac{1}{2}$.

Step4: Calculate P(B')

$P(B') = 1 - P(B)=1-\frac{1}{20}=\frac{19}{20}$.

Step5: Calculate P(A∩B)

Since 11 is odd, $P(A\cap B) = 0$.

Step6: Calculate P(A∪B)

Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, we have $P(A\cup B)=\frac{1}{2}+\frac{1}{20}-0=\frac{10 + 1}{20}=\frac{11}{20}$.

Step1: Calculate P(A)

There are 26 black cards in a standard deck of 52 cards, so $P(A)=\frac{26}{52}=\frac{1}{2}$.

Step2: Calculate P(B)

There are 4 Jacks in a standard deck of 52 cards, so $P(B)=\frac{4}{52}=\frac{1}{13}$.

Step3: Calculate P(A')

$P(A') = 1 - P(A)=1-\frac{1}{2}=\frac{1}{2}$.

Step4: Calculate P(B')

$P(B') = 1 - P(B)=1-\frac{1}{13}=\frac{12}{13}$.

Step5: Calculate P(A∩B)

There are 2 black Jacks (Jack of spades and Jack of clubs), so $P(A\cap B)=\frac{2}{52}=\frac{1}{26}$.

Step6: Calculate P(A∪B)

Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, we have $P(A\cup B)=\frac{1}{2}+\frac{1}{13}-\frac{1}{26}=\frac{13 + 2-1}{26}=\frac{7}{13}$.

Answer:

$P(A)=\frac{1}{4}$, $P(B)=\frac{1}{4}$, $P(A')=\frac{3}{4}$, $P(B')=\frac{3}{4}$, $P(A\cap B)=\frac{1}{10}$, $P(A\cup B)=\frac{2}{5}$

2.