the probability sue purchases a dress is 3/5. if she does buy a dress, the probability she buys shoes is 5/8. If she does not buy a dress, the probability she buys shoes is 1/4. If you learn Sue bought shoes, what is the probability she bought a dress?

Well, let's see. If Sue buys a dress, the probability she buys shoes is 5/8. And if she doesn't buy a dress, the probability she buys shoes is 1/4. So, to calculate the probability she bought a dress given that she bought shoes, we need to consider both scenarios.

First, let's calculate the probability she buys shoes and a dress, which is (3/5) * (5/8) = 15/40.

Now, let's calculate the probability she buys shoes but no dress, which is (2/5) * (1/4) = 2/20.

Adding these two probabilities together, we get 15/40 + 2/20 = 17/40.

So, the probability that Sue bought a dress given that she bought shoes is 17/40.

To find the probability that Sue bought a dress given that she bought shoes, we need to apply Bayes' theorem.

Let's define the following probabilities:
- P(D) = probability of buying a dress = 3/5
- P(D') = probability of not buying a dress = 1 - P(D) = 2/5
- P(S|D) = probability of buying shoes given that a dress was bought = 5/8
- P(S|D') = probability of buying shoes given that a dress was not bought = 1/4

We want to find P(D|S), the probability of buying a dress given that Sue bought shoes.

According to Bayes' theorem, we can calculate this as:

P(D|S) = (P(S|D) * P(D)) / [ (P(S|D) * P(D)) + (P(S|D') * P(D')) ]

Substituting the given values:

P(D|S) = (5/8 * 3/5) / [ (5/8 * 3/5) + (1/4 * 2/5) ]
= 15/40 / [ 15/40 + 2/20 ]
= 15/40 / 17/40
= 15/17

Therefore, the probability that Sue bought a dress given that she bought shoes is 15/17.

To find the probability that Sue bought a dress given that she bought shoes, we need to use conditional probability.

Let's use the following probabilities:
P(D) = probability that Sue purchases a dress = 3/5
P(¬D) = probability that Sue does not purchase a dress = 1 - P(D) = 2/5 (since there are only two possible outcomes: buying a dress or not buying a dress)

Let's also define the following:
P(S|D) = probability that Sue buys shoes given that she purchased a dress = 5/8
P(S|¬D) = probability that Sue buys shoes given that she did not purchase a dress = 1/4

We want to find P(D|S), which is the probability that Sue bought a dress given that she bought shoes.

Using Bayes' Theorem, we have:

P(D|S) = (P(S|D) * P(D)) / P(S)

We need to calculate P(S), which is the probability that Sue bought shoes regardless of whether or not she bought a dress.

P(S) = P(S|D) * P(D) + P(S|¬D) * P(¬D)
= (5/8) * (3/5) + (1/4) * (2/5)
= 15/40 + 2/20
= 3/8

Now we can substitute the values back into Bayes' Theorem to find P(D|S):

P(D|S) = ( (5/8) * (3/5) ) / (3/8)
= ( (5/8) * (3/5) ) / (3/8) * (8/8) (multiplying both numerator and denominator by 8/8 to simplify)
= 15/40 / 24/40
= 15/24
= 5/8

Therefore, the probability that Sue bought a dress given that she bought shoes is 5/8.

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