[Math] Baye’s Theorem Conditional Probability with multiple conditions

bayes-theorembayesianprobability

Lets assume I have a supermarket and I track the purchase history of my customers with 2 attributes of each customer – Gender (M/F) & Smiling (Y/N).

Assume this is historical data of purchases:

     |  Total  |  Male  |  Smile
Beer |     25  |    20  |     22      

Total articles sold including beer:  40
Total number of customers:           60

I need to find the this conditional probability:

Probability that a given person who is male and is smiling will buy a beer.

I need —> P(Beer/Male and Smiling)

Bayes Theroem: 

P(A|B) = P(B|A) * P(A) / P(B)

Applying this to my problem:

P(B|M,S) = P(M,S|B) * P(B) / P(M,S)
         = {P(M|B) * P(S|B)} * P(B) / {P(M) * P(S)}
         = (20/25 * 22/25) * 25/40 / (20/60) * (22/60)
         = 0.70 * 0.625 / 0.122
         = 3.58

I am clearly doing something very, very wrong. Need some guidance.

Assumptions I have made:

  1. Probability of Male and Smiling is independent – I guess this is where the issue lies
  2. P(M,S|B) – This component's calculation & formula – are they right?
  3. Is it an issue with the data?

Edit in response to Guillame's answer:

Let me define those for you:

Assuming that Male & Smiling = 15

  1. Smiling Males bought beer: 10
  2. Smiling Male did not buy beer: 5

Now given this information, where exactly am I going wrong?

Best Answer

With the information you have now you don't really need Bayes. By the basic definition of conditional probability $$ P(B \mid M, S) = \frac{P(M, S, B)}{P(M, S)} = \frac{P(\text{Male, smiling, bought beer})}{P(\text{Male, smiling})} = \frac{10/60}{15/60}= \frac{2}{3} $$

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