# Conditional Probability (Different Context)

A male customer visiting the suit department of a certain store will purchase a suit with probability 0.16, a shirt with probability 0.18, and a tie with probability 0.2. The customer will purchase both a suit and a shirt with probability 0.07, both a shirt and a tie with probability 0.09, and with no probability of purchasing a suit and a tie.

(a) What is the probability that a customer purchases at least one of these items?

P = 0.16 + 0.18 + 0.2 - 0.09 - 0.07 = 0.38

(b) Given that the customer purchases a shirt, what is the probability that he also purchases a tie?

P = 0.09/0.18 = 0.5

(c) Given that the customer purchases at least one item, what is the probability that he purchases a shirt and a tie?

I am confused about this part on conditional probability, as the context is different from part b. Would be very grateful to receive any constructive feedback regarding my question. Thank you!

• What is the probability that the customer buys nothing at all?
– user694818
Aug 24, 2019 at 12:38
• probability will be 1 - P(at least one item) = 1 - 0.38 = 0.62 Aug 24, 2019 at 12:42

(c) would be $$\frac{0.09}{0.38}$$ based on the probability under (a). So about $$0.234$$ or exactly $$\frac{9}{38}$$ (which cannot be simplified).
Or look at it this way: we can draw a Venn diagram (for 100 a costumers, say) for three sets Suits, Shirts and Ties, with thriple intersection having 0, Suit $$\cap$$ Shirts having $$7$$ elements, Suit $$\cap$$ Ties having $$0$$ and Shirts $$\cap$$ Ties having $$9$$ elements. The "unique" part of Shirts has $$2$$ elements ($$18 - 9 - 7$$), the unique part of Suits has $$9$$ ($$16-7$$) and for Ties $$11$$ ($$20 - 9$$). So all disjoint parts together have $$9+7+2+9+11 = 38$$ members buying something, nicely corroborating $$a)$$. There are $$100 - 38 = 62$$ customers outside the three sets. As to $$b)$$, Shirts has $$18$$ members, $$9$$ of which also are in Tie, so we get $$\frac{1}{2}$$.
Now of the $$38$$ buying at least something, $$9$$ buy a shirt and a tie. So given that we pick a customer from inside the Venn diagram, we clearly get $$\frac{9}{38}$$ as the answer.