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 does 'and' give a smaller answer than either event on its own?
- because multiplying two numbers always makes a smaller number than adding them
- because an outcome now has to pass two separate filters, and each one throws some away
- because the second of the two events is always harder to achieve than the first one is
- because probabilities have to be shared out between the two events involved
P(A) = 0.4 and P(B) = 0.5, independent. What is P(A and B)?
Answer: ______________
A fair coin is flipped twice. What is the probability of getting two heads?
Answer: ______________
P(A) = 0.3, P(B) = 0.4, mutually exclusive. What is P(A or B)?
- 0.7
- 0.12
- 0.3
- 1
A fair die is rolled twice. What is P(six then six)?
- 1/12
- 1/6
- 1/18
- 1/36
A stall is won with probability 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: ______________
If a fair coin has landed heads five times running, the next flip is still an even chance.
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 an outcome now has to pass two separate filters, and each one throws some away · Requiring both narrows the set of outcomes that count. Only the ones that satisfy the first and then also satisfy the second survive.
0.2 · 0.4 times 0.5 = 0.2.
0.25 · A coin has no memory, so the flips are independent: 0.5 × 0.5 = 0.25.
0.7 · Exclusive OR adds: 0.3 + 0.4 = 0.7. The 0.12 option wrongly multiplies.
1/36 · One sixth times one sixth = one over 36. The 1 over 12 option wrongly adds.
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 · Flips are independent, so the coin carries nothing forward from the run. Five heads is evidence worth weighing about whether the coin is fair, and it changes nothing about a coin that is.