The definition of independent events is that neither can affect the other: if A and B are independent then the probability of A happening does not depend on whether B has happened or will happen, and vice versa. There is an equation that we can use when two events A and B are independent:
In plain English this says that if two events are independent then to find the probability of them both happening we multiply the individual probabilities together.
The equation may be used in the following way. Suppose John and Bill take their driving tests on the same day. The probability that John will pass is 0.6 and the probability that Bill will pass is 0.3.Find the probability that
c)Exactly one passes
d)At most one passes.
We start by drawing a probability tree:
a)The probability John Passes AND Bill Passes = 0.6*0.3=0.18
b)The prbability That John fails And Bill fails =0.4*0.7=0.28
c) Exactly can pass in two ways:
John can pass AND Bill can fail =0.6*0.7=0.42
John can fail AND Bill can pass =0.4*0.3=0.12
Because either the first OR the second can happen, we add the two probabilities 0.42+0.12=0.54
d)At least one passes mean that both can pass OR exactly one can pass
Example: A bag contains 4 red balls and 7 green balls. Two balls are taken out one at a time and put to one side. Find the probability that
a)Both are red
b)One is red
c)At least one is red
d)both are the same colour
e)Both are different colours
To start we have 4 red balls out of 11, so the probability of picking a red ball isNow we take the ball and put it aside. There are only 3 red balls and 7 red balls out of 10. The probability of the second ball being red isand the probability of the second ball being green isThis labels the top half of the probability tree as shown.
The probability of the first ball being green isThen this ball is put aside and there are now 4 red balls and 6 green balls out of 10, so the probability of the second ball being red isand the probability of the second ball being green isThis labels the bottom half of the probability tree.
a)First Ball Red and Second Ball Red=
b)One of the two balls can be red in two ways:
The first ball is red and the second one is green =
The first ball can be green and the second ball can be red =
Since we can have either the first OR the second way round we add these two answers:
C)At least one is read mean both can be red, OR exactly one can be red, so we add the answers to a) and b)
d)Both have the same colour if they are both red OR if they are both green
The probability of the first one being green and second one being green is
Now we add this and the answer to a)
d) If they are different colours, they cannot be the same colour so we find 1 minus the answer to d) =