- CalculatorCalculatorXMRMCM+()789÷C456×CE123−√±0.+=Transfer Display
- Events A and B are independent.
- The probability that events A and B both occur is 0.6
|Column A||Column B|
|The probability that event A occurs||0.3|
Prob A and B equals 0point6
This problem is testing your knowledge of independent events, which is a specific concept in probability. If A and B are independent, then P(A & B) = P(A)
P(B). At Magoosh we usually say not to memorize formulas because it's better to learn how to use them. This is one you'll probably need to memorize :) Once you do, it's pretty easy. If the probability of both events occurring is 0.6, we're most of the way to the answer. Both events occurring is "and," meaning A happens AND B happens. You should remember that "and" means multiply and "or" means add. So, The probability of both events occurring is 0.6, and we get that by multiplying P(A) and P(B). Finally, remember that when you multiply decimals or fractions (anything between 0 and 1), the result will be smaller. That means that both P(A) and P(B) need to be larger than 0.6 in order to equal 0.6 when multiplied. It makes sense if you think about it. 0.6
0.6 = 0.36. Too small! What about smaller numbers? 0.5
0.5 = 0.25, definitely too small. Larger numbers would do it though! 0.775
0.775 = 0.6. Of course we don't know that P(A) and P(B) are equal. This is just a demonstration of why we need numbers larger than 0.6. Of course, a number larger than 0.6 is larger than 0.3, so Column A is larger, and (A) is the answer.
FAQ: I am still confused about why you can conclude that P(A)> = 0.6 and P(B)> = 0.6?
A: The only thing we know for sure is that the probability of both A and B is 0.6. We can then take the equation P(A & B) = P(A) x P(B). So again, we know that P(A & B) = 0.6. I think the best way to explain this would be to plug in numbers and see what we get.
We know that the probability of something cannot be greater than 1, so P(A) < 1. We also know that the probability of A must be greater than zero, because P(A & B)= 0.6. So, we know 0 < P(A) < 1. Now let's plug in some numbers.
If, for example, P(A) = 0.4, we would have the equation 0.6 = 0.4 x P(B). This would make P(B) = 1.5, which we know isn't possible. And if P(A) = 0.5? P(B) would equal 1.2 (again, not possible). Let's try 0.6. 0.6 = 0.6 x P(B). P(B) would equal 1. That's possible! So, now we know that the probability of both A and B must be greater than 0.6, otherwise the other probability will be greater than 1.
Thus, the probability that event A occurs is greater than 0.6, and therefore greater than the quantity in Column B (0.3).
Watch the lessons below for more detailed explanations of the concepts tested in this question. And don't worry, you'll be able to return to this answer from the lesson page.