به انگلیسی

When two things both have to happen

Mathematics · Probability · ages 15-16

Name ______________________ Date ____________

Two independent events have probabilities 0.3 and 0.4. What is the probability that both happen?
Answer: ______________
Why can P(A and B) never be greater than either P(A) or P(B)?
- because multiplying any two positive numbers gives less than either number
- because only outcomes belonging to both events count
- because the second event always has a lower probability than the first
- because independent events always have probabilities that add up to one
P(A) = 0.6 and P(B) = 0.5, and A and B are independent. What is P(A and B)?
Answer: ______________
A fair coin is flipped twice, independently. What is the probability of getting two heads?
Answer: ______________
A and B are mutually exclusive, with P(A) = 0.3 and P(B) = 0.4. What is P(A or B)?
- 0.7
- 0.12
- 0.3
- 1
A fair six-sided die numbered 1 to 6 is rolled twice, independently. What is P(six then six)?
- 1/3
- 1/6
- 1/18
- 1/36
The chance of winning at a stall is 0.3. Three friends each have one go, independently of each other. What is the probability that all three win?
Answer: ______________
How would you check whether two events are independent, given the numbers?
- see whether the two probabilities happen to be equal to one another exactly
- see whether they add up to 1 when put together
- see whether the two events are able to both happen at the same time
- see whether the second's probability is the same once the first is known
A and B are independent. P(A) = 0.4 and P(A and B) = 0.12. What is P(B)?
Answer: ______________
A fair coin is flipped independently each time. It has landed heads five times running. True or false: the next flip still has an even chance of heads.
Circle one: True False

Answer key

For grown-ups. Fold this page away before handing over the rest.

When two things both have to happen W1-mt_MK8EPHEUdN-s1

0.12 · For independent events the probabilities multiply: 0.3 × 0.4 = 0.12.
because only outcomes belonging to both events count · The outcomes in both events are contained within each individual event. Requiring both can remove outcomes or keep them all, but it cannot add any.
0.3 · Independent events multiply: 0.6 × 0.5 = 0.3.
0.25 · The question gives independent flips. Each flip has a 0.5 chance of heads, so 0.5 × 0.5 = 0.25.
0.7 · For mutually exclusive events, or means add: 0.3 + 0.4 = 0.7. The 0.12 option wrongly multiplies.
1/36 · The rolls are independent, as stated, so multiply: 1/6 × 1/6 = 1/36. Adding gives 1/3, while 1/6 is the chance of a six on only one roll.
0.027 · Independence lets all three multiply: 0.3 × 0.3 × 0.3 = 0.027.
see whether the second's probability is the same once the first is known · Independence is about information. Work out the second event's probability given that the first happened, and compare it with the second's probability on its own.
0.3 · Independence gives 0.4 × P(B) = 0.12, so P(B) = 0.12 / 0.4 = 0.3.
True · The stated independence leaves the next heads chance at 0.5. Five heads is evidence to consider when judging fairness, but here fairness and independence are given.