Find the conditional probability that a randomly selected male and randomly selected female earn 50 or more
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Answer:
In probability, we say two events are independent if knowing one event occurred doesn't change the probability of the other event.
For example, the probability that a fair coin shows "heads" after being flipped is 1/21/21, slash, 2. What if we knew the day was Tuesday? Does this change the probability of getting "heads?" Of course not. The probability of getting "heads," given that it's a Tuesday, is still 1/21/21, slash, 2. So the result of a coin flip and the day being Tuesday are independent events; knowing it was a Tuesday didn't change the probability of getting "heads."
Not every situation is this obvious. What about gender and handedness (left handed vs. right handed)? It may seem like a person's gender and whether or not they are left-handed are totally independent events. When we look at probabilities though, we see that about 10\%10%10, percent of all people are left-handed, but about 12\%12%12, percent of males are left-handed. So these events are not independent, since knowing a random person is a male increases the probability that they are left-handed.
The big idea is that we check for independence with probabilities.
Two events, A and B, are independent if P(\text{A } | \text{ B})=P(\text{A})P(A ∣ B)=P(A)P, left parenthesis, start text, A, space, end text, vertical bar, start text, space, B, end text, right parenthesis, equals, P, left parenthesis, start text, A, end text, right parenthesis and P(\text{B } | \text{ A})=P(\text{B})P(B ∣ A)=P(B)P, left parenthesis, start text, B, space, end text, vertical bar, start text, space, A, end text, right parenthesis, equals, P, left parenthesis, start text, B, end text, right parenthesis.