Video Transcript
Suppose that 𝐴 and 𝐵 are events
in a random experiment. Given that the probability of 𝐴 is
0.71, the probability of 𝐵 bar is 0.47, and the probability of 𝐴 union 𝐵 is 0.99,
determine the probability of 𝐵 minus 𝐴.
Before starting this question, we
recall that 𝐵 bar means the complement of event 𝐵. The probability of the complement
is the same as the probability of the event not occurring. As probabilities sum to one, we
know the probability of 𝐵 bar is equal to one minus the probability of 𝐵. Rearranging this formula, the
probability of event 𝐵 is therefore equal to one minus the probability of the
complement of 𝐵. As this is equal to 0.47, we can
subtract this from one to calculate the probability of event 𝐵. The probability of event 𝐵 is
therefore equal to 0.53. The reason we need this value is we
are asked to calculate the probability of 𝐵 minus 𝐴. Recalling the difference rule for
probability, we know this is equal to the probability of 𝐵 minus the probability of
𝐴 intersection 𝐵.
We now know that the probability of
𝐵 is 0.53, and we can calculate the probability of 𝐴 intersection 𝐵. We can do this using the addition
rule of probability, one form of which states that the probability of 𝐴
intersection 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus
the probability of 𝐴 union 𝐵. We are told in the question that
the probability of 𝐴 is 0.71. We have calculated that the
probability of 𝐵 is 0.53, and we are also given that the probability of 𝐴 union 𝐵
is 0.99. The probability of 𝐴 intersection
𝐵 is therefore equal to 0.71 plus 0.53 minus 0.99. This is equal to 0.25. Substituting this value into the
difference rule for probability, we see that the probability of 𝐵 minus 𝐴 is equal
to 0.53 minus 0.25. This gives us a final answer of
0.28.
