Consider a large box containing bags of tulip bulbs. Each bag contains 25 tulip

Question

Consider a large box containing bags of tulip bulbs. Each bag contains 25 tulip bulbs. Three–fourths of the bags contain bulbs for five red and 20 yellow tulips; one–fourth of the bags contain bulbs for 15 red and 10 yellow tulips. A bag is selected at random and one bulb, selected at random from this bag, is planted.

a) Find the probability that this bulb will produce a red tulip.
b) Find the probability that this bulb will produce a yellow tulip.

c) Suppose that the bulb produces a red tulip. Find the conditional probability that the bulb came from a bag containing bulbs for 15 red and 10 yellow tulips.

Explanation / Answer

Let Bag 1 contain (5R 20Y) and Bag 2 containn (15R, 10Y)

P(B1) = 0.75, P(B2) = 0.25 (R from B1) = 5/25 = 0.2 and P(Y from B1) = 20/25 = 0.8, P(R from B2) = 15/25 = 0.6 and P(Y from B2) = 10/25 = 0.4

a) Find the probability that this bulb will produce a red tulip.

P(Bag1)P(R/Bag1)=0.75 × 5/25 = 0.75 × 0.2 = 0.15

P(Bag2)P(R/Bag2)=0.25 ×15/25 = 0.25 × 0.6 = 0.15

Therefore the required probability = P(R) = 0.15 + 0.15 = 0.3

b) Find the probability that this bulb will produce a yellow tulip.

P(Bag1)P(Y/Bag1) = 0.75 × 20/25 = 0.75×0.8 = 0.6

P(Bag2)P(Y/Bag2) = 0.25 ×10/25 = 0.25 × 0.4 = 0.1

Therefore the required probability = P(Y)= 0.6 + 0.1 = 0.7

c) Suppose that the bulb produces a red tulip. Find the conditional probability that the bulb came from a bag containing bulbs for 15 red and 10 yellow tulips.

Here we need to find the probability that given a red tulip is produced, it came from Bag2 i.e

P(B2/R) =[ P(R/B2) * P(B2) ]/ P(R)

P(R/B2) = Probability of getting a red tulip given it is Bag 2 = 15/25 = 0.6

P(Bag2) = 0.25 and P(R) is what we have calculated in (a) = 0.3

Therefore P(B2/R) = 0.6*0.25/0.3 = 0.5

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