Probability Explanation And Exercises Worksheet Page 10
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37method, it has the advantage of being applicable to problems with more than two
events. Here is the calculation in the present case. The probability of not getting
either a 6 or a head can be recast as the probability of
(not getting a 6) AND (not getting a head).
This follows because if you did not get a 6 and you did not get a head, then you did
not get a 6 or a head. The probability of not getting a six is 1 - 1/6 = 5/6. The
probability of not getting a head is 1 - 1/2 = 1/2. The probability of not getting a six
and not getting a head is 5/6 x 1/2 = 5/12. This is therefore the probability of not
getting a 6 or a head. The probability of getting a six or a head is therefore (once
again) 1 - 5/12 = 7/12.
If you throw a die three times, what is the probability that one or more of
your throws will come up with a 1? That is, what is the probability of getting a 1
on the first throw OR a 1 on the second throw OR a 1 on the third throw? The
easiest way to approach this problem is to compute the probability of
NOT getting a 1 on the first throw
AND not getting a 1 on the second throw
AND not getting a 1 on the third throw.
The answer will be 1 minus this probability. The probability of not getting a 1 on
any of the three throws is 5/6 x 5/6 x 5/6 = 125/216. Therefore, the probability of
getting a 1 on at least one of the throws is 1 - 125/216 = 91/216.
Conditional Probabilities
Often it is required to compute the probability of an event given that another event
has occurred. For example, what is the probability that two cards drawn at random
from a deck of playing cards will both be aces? It might seem that you could use
the formula for the probability of two independent events and simply multiply 4/52
x 4/52 = 1/169. This would be incorrect, however, because the two events are not
independent. If the first card drawn is an ace, then the probability that the second
card is also an ace would be lower because there would only be three aces left in
the deck.
Once the first card chosen is an ace, the probability that the second card
chosen is also an ace is called the conditional probability of drawing an ace. In this
case, the “condition” is that the first card is an ace. Symbolically, we write this as:
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