Probability question?

Question

suppose that of all individuals purchasing a digital camera 60% include an optional memory card, 40% include an extra battery and 30% include both battery and memory card.

a. what is the probability that a person purchased an extra card if we know that they purchased an extra battery?


b. what is the probability that the person purchased an extra battery if we know that they purchased an extra memory card?


I'm not sure what formulas to use here

would I use P(A or B) = P(A) + P(B) - P(AB)
or P(A|B) = P (AB) / P(B)

or some other ?

Thank you!

Answer 1

I've always had a hard time with the formulas. This is much easier with Venn diagrams. Make 2 circles that overlap. Label one memory, the other battery. In the overlap, where you have BOTH memory and battery, write 30. In the other part of the memory circle write 30 (the rest of the 60), and in the other part of the battery circle write 10 (the rest of the 40).

Now the questions are easy.

a. If we know they bought an extra battery, 30/40 of them also bought a memory card.

b. If we know they bought a memory card, 30/60 of them also bought a battery.

Look at the numbers now, and the formulas:

P(mem or bat) = 70%. That's P(mem) + P(bat) - P(both) = 60 + 40 - 30.

P(mem | bat) = 30/40 = P(both) / P(bat) and
P(bat | mem) = 30/60 = P(both) / P(mem)

Answer 2

We are assuming the probabilities are NOT independent. They fall in 4 separate categories - nothing, card, battery, both. However, I think the trick is, based on wording, that the "60% card" includes the "30% card and battery". Ditto, 40% battery includes the 30% c&b.

SO the actual distributions are: (subtraction)
Pcb = 30% C&B (given)
Pco = 30% Card-only (60-30) (Pc-Pcb)
Pbo = 10% Battery Only (40-30) (Pb - Pcb)
Pnada = Remainder 30% nothing extra. (100-30-30-10)

(a) So Pb = 40, Pcb = 30; 40% of total bought a battery, 30% of total also bought a card. 30/40 = 75% of battery people bought a card too.

(b) Similarly, 60% of total bought a card. 30% of total bought a card and battery. 30/60 = 50% of card people bought a battery too.