# Video: GCSE Mathematics Foundation Tier Pack 4 • Paper 2 • Question 22

GCSE Mathematics Foundation Tier Pack 4 • Paper 2 • Question 22

02:27

### Video Transcript

Spencer is going clay pigeon shooting. Below is a probability tree diagram for whether he hits or misses the first two targets. Work out the probability that Spencer misses both targets.

Here we have a probability tree diagram that is fully complete. There is a little trick to help us decide what to do with the probabilities to answer any question. We’re going to write a little note at the end of each pathway to show what’s happened.

Up here, Spencer has hit target one and target two. We can write that as HH, standing for hit, hit. Here Spencer has hit the first target and missed the second. That’s HM, standing for hit, miss. Here Spencer’s missed the first target and hit the second. That’s MH. And down here, he’s missed both targets. That’s miss, miss.

By writing the outcomes like this, we’ve reminded a little bit of algebra. In algebra, when two letters are next to each other, they are multiplying one another. This reminds us that, to find the probabilities for each outcome, we can multiply their respective probabilities. In fact, we are only interested in the probability that Spencer misses both targets. That’s this one.

To find the probability he misses both targets then, we multiply the probability he misses the first target by the probability he misses the second. That’s 0.8 multiplied by 0.6. To perform this multiplication, we can first multiply eight by six to give us 48.

Since each number in our calculation is actually 10 times smaller than those in the calculation we performed, we divide 48 by 10 and then by 10 again. Dividing by 10 gives us 4.8. Then dividing by 10 again, we get 0.48. The probability that Spencer misses both targets then is 0.48.

As a little side note, the rule we use is sometimes called the “AND rule.” When two events, 𝐴 and 𝐵, are independent — that’s to say that the outcome of one event doesn’t affect the outcome of the other — we can find the probability of 𝐴 and 𝐵 occurring by multiplying their probabilities. As you can see, that’s 0.8 multiplied by 0.6 for the probability that he misses target one and the probability he misses target two. We have shown that the probability he misses both targets is 0.48.