Question

The letters of the word product are arranged in all possible ways. If an arrangement is picked at random. What is the probability that the arrangement will start with p and ends with a vowel?

Answer 1

It is given that the letters of the word product are arranged in all possible ways. If an arrangement is picked at random we have to find the probability that the arrangement will start with p and ends with a vowel.

We know, Probability of any event = (Number of favorable outcomes)/(Total number of possible outcomes).

Our given word is "product". There are 7 letters in this word. Thus, number of ways for arranging these 7 letters = 7! = 7*6*5*4*3*2*1 = 5040

Thus, total number of possible outcomes = 5040.

Now, in the word "product", there is one p. Thus, the number of arrangements in which the word will start with p = 1. ----------(1)

Now, there are 2 vowels (o,u) in the word "product". Thus, number of arrangements in which the word will end with a vowel = 2. -------(2)

Now, after arranging the starting and ending letters, there are 5 letters left for the middle. Thus, these 5 letters can be arranged in 5! ways. Thus, the middle letters can be arranged in 5! ways = 5*4*3*2*1 = 120. ------------(3)

Thus, number of ways of arranging the letters = 1*2*120 = 240 [From (1), (2), (3)]

Thus, number of favorable outcomes = 240.

Thus, the required probability = 240/5040 = 0.0476(rounded up to four decimal places).

Thus, the probability that the arrangement will start with p and ends with a vowel = 0.0476.