Question

Each bag in a large box contains 25 tulip bulbs. It is known that \(60 \%\) of the bags contain bulbs for 5 red and 20 yellow tulips, while the remaining \(40 \%\) of the bags contain bulbs for 15 red and 10 yellow tulips. A bag is selected at random and a bulb taken at random from this bag is planted. (a) What is the probability that it will be a yellow tulip? (b) Given that it is yellow, what is the conditional probability it comes from a bag that contained 5 red and 20 yellow bulbs?

Short answer

The probability that a randomly planted bulb will be a yellow tulip is \(0.56\) (or \(56\% \)). Given that a bulb is yellow, the conditional probability that it comes from a bag that contained 5 red and 20 yellow bulbs is \(0.8571\) (or \(85.71\% \)).

Step-by-step solution

Set up and calculate all unconditioned probabilities

Firstly, the probability of picking a bag with 5 red and 20 yellow tulips is \(P(Bag_1) = 0.6\) and the probability of picking a bag with 15 red and 10 yellow tulips is \(P(Bag_2) = 0.4\). Then, the probability of picking randomly a yellow tulip from Bag_1 is \(P(Y|Bag_1) = 20/25 = 0.8\) and from Bag_2 is \(P(Y|Bag_2) = 10/25 = 0.4\).

Calculate total probability of picking a yellow tulip

Using the total probability theorem, calculate the total probability of selecting a yellow tulip bulb which includes the probability of selecting a yellow tulip from each type of bag and multiplying it by the probability of selecting that bag itself. Thus, \(P(Y) = P(Y \cap Bag_1) + P(Y \cap Bag_2) = P(Bag_1) \cdot P(Y|Bag_1) + P(Bag_2) \cdot P(Y|Bag_2) = 0.6 \cdot 0.8 + 0.4 \cdot 0.4 = 0.56.\)

Use Bayes' Theorem to solve for conditional probability

Given that the bulb is yellow, use Bayes' Theorem to find out what is the conditional probability it comes from a bag that contained 5 red and 20 yellow bulbs. According to the Bayes' Theorem, \(P(Bag_1 | Y) = \frac{P(Y | Bag_1) \cdot P(Bag_1)}{P(Y)} = \frac{0.8 \cdot 0.6}{0.56} = 0.8571.\)