# Video: Determining the Probability of Difference between Two Events

Suppose 𝐴 and 𝐵 are two events. Given that 𝑃(𝐴) = 0.3 and 𝑃(𝐴 ∩ 𝐵) = 0.03, determine 𝑃(𝐴 − 𝐵).

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### Video Transcript

Suppose 𝐴 and 𝐵 are two events. Given that the probability of 𝐴 is equal to 0.3 and the probability of 𝐴 intersection 𝐵 is 0.03, determine the probability of 𝐴 and not 𝐵.

If we think about these events in a Venn diagram, the probability of 𝐴 intersection 𝐵 would be this space, the space for 𝐴 which does not include the intersection. This is the space when only event 𝐴 occurs. Another way to say that would be event 𝐴 minus event 𝐵. Since we know that the intersection of 𝐴 and 𝐵 is 0.03, we can make the statement that the probability of 𝐴 minus 𝐵 plus the probability of 𝐴 intersection 𝐵 will be equal to the whole probability of 𝐴.

This means the probability of 𝐴 minus 𝐵 plus 0.03 must be equal to 0.3. If we subtract 0.03 from both sides of our equation, we get that the probability of 𝐴 minus 𝐵 will be equal to 0.27. This makes sense because 0.27 plus 0.03 does equal 0.3.

The probability of 𝐴 minus 𝐵 is the probability that only event 𝐴 occurs and event 𝐵 does not occur, and under these conditions that would be 0.27.