Statistics: Probability Rules: Addition and Multiplication
Using addition, multiplication, and conditional probability rules
Statistics: Probability Rules: Addition and Multiplication
Using addition, multiplication, and conditional probability rules
Statistics - Grade 9-12
- 1
In a class, P(student plays a sport) = 0.45, P(student plays an instrument) = 0.30, and P(student does both) = 0.12. Find the probability that a randomly chosen student plays a sport or an instrument.
For overlapping events, subtract the probability of both events so it is not counted twice.
The probability is 0.63. Use the addition rule: P(sport or instrument) = 0.45 + 0.30 - 0.12 = 0.63. - 2
A spinner has 8 equal sections numbered 1 through 8. Find the probability of spinning an even number or a number greater than 6.
The probability is 5/8. The even numbers are 2, 4, 6, and 8, and the numbers greater than 6 are 7 and 8. There are 5 outcomes in the union: 2, 4, 6, 7, and 8. - 3
Two events A and B are mutually exclusive. P(A) = 0.28 and P(B) = 0.35. Find P(A or B).
Mutually exclusive events cannot happen at the same time.
The probability is 0.63. Since A and B are mutually exclusive, P(A and B) = 0, so P(A or B) = 0.28 + 0.35 = 0.63. - 4
A card is drawn from a standard 52-card deck. Find the probability that the card is a king or a heart.
The probability is 16/52, which simplifies to 4/13. There are 4 kings and 13 hearts, but the king of hearts is counted in both groups, so 4 + 13 - 1 = 16 favorable cards. - 5
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. One marble is chosen, replaced, and then a second marble is chosen. Find the probability that both marbles are red.
With replacement, the total number of marbles stays the same for each draw.
The probability is 1/4. Since the marble is replaced, the events are independent, so P(red and red) = 5/10 × 5/10 = 25/100 = 1/4. - 6
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. One marble is chosen and not replaced, then a second marble is chosen. Find the probability that both marbles are red.
The probability is 2/9. Without replacement, the events are dependent, so P(red and red) = 5/10 × 4/9 = 20/90 = 2/9. - 7
Events A and B are independent. P(A) = 0.60 and P(B) = 0.25. Find P(A and B).
Independent means one event does not change the probability of the other event.
The probability is 0.15. For independent events, P(A and B) = P(A) × P(B), so 0.60 × 0.25 = 0.15. - 8
A student is selected at random from a school. P(student is a senior) = 0.22. Among seniors, P(student has a parking permit) = 0.70. Find the probability that the selected student is a senior and has a parking permit.
The probability is 0.154. Use the multiplication rule with conditional probability: P(senior and permit) = P(senior) × P(permit given senior) = 0.22 × 0.70 = 0.154. - 9
In a survey, 40% of students like math, 50% like science, and 20% like both math and science. Find the probability that a randomly selected student likes neither math nor science.
Find the probability of at least one first, then subtract from 1.
The probability is 0.30. First find P(math or science) = 0.40 + 0.50 - 0.20 = 0.70. Then use the complement: P(neither) = 1 - 0.70 = 0.30. - 10
A fair six-sided die is rolled twice. Find the probability of rolling a 6 on the first roll and an odd number on the second roll.
The probability is 1/12. The rolls are independent, so P(6 then odd) = 1/6 × 3/6 = 3/36 = 1/12. - 11
A two-way table shows 120 students by grade level and whether they ride the bus. There are 50 freshmen, 70 sophomores, 35 freshmen who ride the bus, and 30 sophomores who ride the bus. Find the probability that a randomly chosen student is a freshman or rides the bus.
First find the total number of bus riders, then use the addition rule.
The probability is 80/120, which simplifies to 2/3. There are 50 freshmen and 65 bus riders total, with 35 students counted in both groups, so 50 + 65 - 35 = 80 students are freshmen or bus riders. - 12
For two events A and B, P(A) = 0.55, P(B) = 0.40, and P(A or B) = 0.75. Find P(A and B), then state whether A and B are mutually exclusive.
P(A and B) is 0.20. Rearranging the addition rule gives P(A and B) = 0.55 + 0.40 - 0.75 = 0.20. The events are not mutually exclusive because P(A and B) is not 0.