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.