Empirical Probability questions?
I have two questions I need help solving:
Problem 1:
In a jar of marbles, there are 17 green, 23 red, 57 blue, and 13 white.
What is the empirical probability of getting a blue marble, dropping it back into the jar, then getting a red marble?
Problem 2:
A family has two children.
What is the empirical probability that the oldest is a boy?
What is the empirical probability that they have a boy and a girl (in any order)?
What is the empirical probability that they have a boy and a girl, with the boy the oldest?
Thank you!!! :)
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Verified answer
Problem 1:
P{drawing a blue marble} = 57 / (17 + 23 + 57 + 13) = 57/120
P{drawing a red marble} = 23 / (17 + 23 + 57 + 13) = 23/120
Note that the denominator counts 57 for blue since we are putting it back into the jar! If we were not putting it back in the jar, it would be a 56 in the denominator and the probability would be 23/119.
Anyways, since we been both events to occur, we want the first AND the second... In probability, AND means we multiply and OR means we add. so here we multiply!
(57/120) x (23/120) = 1311/14400 = 0.091
So there's about a 9.1% chance that you'll draw twice and get a blue and a red.
Problem 2:
Assume the likelihood of a boy and a girl is equal (i.e., P{ boy } = P{ girl } = 1/2)
P{ oldest is a boy } = 1/2,
"Boy and a girl in any order", this occurs when {1st is boy AND 2nd is girl} OR {1st is a girl AND 2nd is a boy}
So,
P{ boy and a girl in any order } = P{1st is boy AND 2nd is girl} + P{1st is a girl AND 2nd is a boy}
P{ boy and a girl in any order } = (1/2) x (1/2) + (1/2) x (1/2) = (1/4) + (1/4) = 1/2
"boy and girl with boy 1st", this occurs when {1st is a boy} AND {2nd is a girl}
P{ boy and girl with boy 1st } = P{ 1st is boy} x P{2nd is girl}
P{ boy and girl with boy 1st } = 1/2 x 1/2 = 1/4