Question

events A and B in some sample sapce each have a probability 0.5 of occuring, while the probability that both occur (simultaneously) is 0.25. what is the probability that neither of A or B occurs? this is the work i did, can i please get some guidance

Answer 1

Answer: 0.25

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Step-by-step explanation:

I appreciate you showing your work so far, and your thought process.

Your steps are correct because the events
`A \cup B` and
`(A \cup B)'` are complementary, aka opposite. One or the other must happen. It's like a coin flip heads vs tails.

The last thing you need to do really is simplify 1-0.5-0.5+0.25 to get the final answer 0.25

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This is what I would do as an approach.

P(A u B) = P(A) + P(B) - P(A n B)

P(A u B) = 0.5 + 0.5 - 0.25

P(A u B) = 0.75

So,

P( (A u B)' ) = 1 - P(A u B)

P( (A u B)' ) = 1 - 0.75

P( (A u B)' ) = 0.25

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Another approach:

Imagine we had 100 people.

50 are in set A to give P(A) = 50/100 = 0.50

50 are in set B to give P(B) = 0.50

25 are in both A and B to get P(A n B) = 25/100 = 0.25

50+50-25 = 75 people are in either A, or B, or both

That leaves 100-75 = 25 in neither set

25/100 = 0.25

I recommend to draw out the Venn Diagram.