Bayes’ formula Principles of Counting

Prior probability: probability before the arrival of any new information

Posterior probability: probability that reflects or comes after the new information

[Practice Problems] A parking lot has 100 red and blue cars in it.

40% of the cars are red.

70% of the red cars have radios.

80% of the blue cars have radios.

What is the probability that the car is red given that it has a radio?

A. 28%.        B. 47%.        C. 37%.

[Solutions] C

In this case, P(red car has a radio) = 0.70 is divided by 0.76 (which is the Unconditional Probability of a car having a radio (40% are red of which 70% have radios) plus (60% are blue of which 80% have radios) or ((0.40) × (0.70)) + ((0.60) × (0.80)) = 0.76.) This result is then multiplied by the Prior Probability of a car being red, 0.40. The result is (0.70 / 0.76) × (0.40) = 0.37 or 37%.

A mutual fund guide ranked 10 mutual funds by total returns for the year 2014. The guide also assigned each fund one of four performance labels: 2-star (3 funds), 3-star (4 funds), 4-star (4 funds), 5-star (3 funds), all the funds are accounted for. How many different ways can we take 10 mutual funds and label three of them 2-star, four 3-star, four 4-star, and 3 5-star, so that each fund is labeled?

[Practice Problems] From an approved list of 25 funds, a portfolio manager wants to rank 4 mutual

funds from most recommended to least recommended. Which formula is most appropriate to calculate the number of possible ways the funds could be ranked?

A.       Permutation formula

B.        Multinomial formula

C.        Combination formula

[Solutions] A

The permutation formula is used to choose r objects from a total of n objects when order matters. Because the portfolio manager is trying to rank the four funds from most recommended to least recommended, the order of the funds matters; therefore, the permutation formula is most appropriate.