How can I combine two probabilities?

Question

Could you please explain using this example? I know it's not true, I made it up and can't think of anything better. I need to figure out for class if arguments are cogent (more likely than not) and it would help a lot if I could do the numbers.

85% of people like chocolate.

35% of people who like vanilla like chocolate.

Jane Doe likes vanilla.

What is the probability that Jane Doe likes chocolate?

Answer 1

Combining them would make 0.85 * 0.35 = 0.2975 which is 29.75%.

But in your question "What is the probability that Jane Doe likes chocolate?" the answer is simply 35%, since the other probability doesn't apply here.

Answer 2

If the two probabilities are independent, you just multiply them to get the probability of both events occurring.

That is not the case in your example. If 85% of people like chocolate, but only 35% of people who like vanilla like chocolate, then they're not independent; a person who likes vanilla is considerably less likely to like chocolate than a person chosen at random (and therefore a person who does NOT like vanilla is more likely to like chocolate).

Since we are given that Jane Doe likes vanilla, and that 35% of people who like vanilla like chocolate, 35% is the answer here; the probability that a random person likes chocolate is irrelevant because we know that Jane's liking vanilla means it doesn't apply.

The one other calculation I can get out of this is that, since 65% of people who like vanilla don't like chocolate and only 15% of all people don't like chocolate, AT MOST just over 23% (15/65) of people like vanilla.

Answer 3

Let A be the event likes chocolate

Let B be the event likes vanilla

We are not assuming independence of events, that is draw two circles which overlap to some extent. Circle A  and Circle B.

The probability that they either like chocolate or vanilla is not

Prob(A) + Prob(B) since the overlap would count the group AandB twice.

So the formula:

Prob(A or B) = Prob(A) + Prob(B) - Prob(AandB)

But your question relates to conditional probabilities

The statement 35% of people who like vanilla like chocolate is written: Prob(A|B), the probablity of A given B is true.

that formula is Prob(AandB)/Prob(B).

You give us Prob(A|B) = 0.35  and that is the answer you are looking for.

Can you see on the circle diagrams where the ratio of these fractions come from?

Answer 4

35%

I don't see what you mean by "combining two probabilities" in this case.