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Chapter 4: Problem 28

An investor has the option of investing in three of five recommended stocks. Unknown to her, only two will show a substantial profit within the next 5 years. If she selects the three stocks at random (giving every combination of three stocks an equal chance of selection), what is the probability that she selects the two profitable stocks? What is the probability that she selects only one of the two profitable stocks?

Short Answer

Expert verified
Answer: The probability that the investor selects the two profitable stocks is 0.3, and the probability that she selects only one of the two profitable stocks is 0.6.

Step by step solution

01

Calculate the total number of ways the investor can select three stocks

From the problem, the investor has to choose three stocks from a total of five stocks. To calculate the total number of possible combinations, we can use the combination formula. C(n, k) = n! / (k!(n-k)!) So, the total number of ways to select three stocks from the given five stocks is: C(5, 3) = 5! / (3!(5-3)!) = 10
02

Calculate the number of ways the investor can select the two profitable stocks

Since there are two profitable stocks among the five recommended stocks, we want to know the number of ways the investor can select those two stocks. The third stock selected must be one of the three remaining non-profitable stocks. The number of ways to select the two profitable stocks is: C(2, 2) = 2! / (2!(2-2)!) = 1 The number of ways to select the third non-profitable stock is: C(3, 1) = 3! / (1!(3-1)!) = 3 Now, we multiply both results to obtain the total number of ways the investor can select the two profitable stocks and one non-profitable stock: 1 * 3 = 3
03

Calculate the probability of selecting the two profitable stocks

To calculate the probability that the investor selects the two profitable stocks, we divide the number of ways the investor can select the two profitable stocks (calculated in Step 2) by the total number of ways to select three stocks from the given five stocks (calculated in Step 1). Probability of selecting the two profitable stocks = 3 (number of ways to select two profitable stocks) / 10 (total number of ways to select three stocks) = 3/10 = 0.3
04

Calculate the number of ways the investor can select only one profitable stock

Now we want to find the number of ways the investor can select only one of the two profitable stocks. To do this, we calculate the number of ways to select one profitable stock from the two, which is: C(2, 1) = 2! / (1!(2-1)!) = 2 Then, we need to calculate the number of ways to select the remaining two non-profitable stocks from the three non-profitable stocks: C(3, 2) = 3! / (2!(3-2)!) = 3 Now, we multiply both results to obtain the total number of ways the investor can select one profitable stock and two non-profitable stocks: 2 * 3 = 6
05

Calculate the probability of selecting only one profitable stock

To calculate the probability that the investor selects only one of the two profitable stocks, we divide the number of ways the investor can select only one profitable stock (calculated in Step 4) by the total number of ways to select three stocks from the given five stocks (calculated in Step 1). Probability of selecting only one profitable stock = 6 (number of ways to select one profitable stock) / 10 (total number of ways to select three stocks) = 6/10 = 0.6 So the probability that the investor selects the two profitable stocks is 0.3, and the probability that she selects only one of the two profitable stocks is 0.6.

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