Question

A corporation is evaluating a proposed merger. The board of directors surveyed 53 stockholders concerning their position on the merger. The results are reported below: (Round the final answers to 1 decimal place.)

Opinion
Number of Shares Held Favour Opposed Undecided Total
Under 200 9 5 3 17
200 to under 1000 4 9 4 17
1000 or more 6 11 2 19
Total 19 25 9 53
a. What is the probability that a stockholder holds 1000 or more shares?

Probability %

b. What is the probability that a stockholder holds less than 200 shares and is in favour of the merger?

Probability %

c. What is the probability that a stockholder holds between 200 to under 1000 shares or is opposed to the merger?

Probability %

d. What is the probability that a stockholder holds under 200 shares given that he/she is opposed to the merger?

Probability %

Answer 1

The probabilities are 35.8% for holding 1000 or more shares, 17.0% for holding less than 200 shares and favoring the merger, 62.3% for holding 200 to under 1000 shares or opposed to the merger, and 20.0% for holding under 200 shares given being opposed to the merger.

Step-by-step explanation:

The student asks for help with calculating probabilities for a corporation's stockholders' opinions on a proposed merger. Here are the answers:

• a. The probability that a stockholder holds 1000 or more shares is 19/53 or approximately 35.8%.
• b. The probability that a stockholder holds less than 200 shares and is in favour of the merger is 9/53 or approximately 17.0%.
• c. The probability that a stockholder holds 200 to under 1000 shares or is opposed to the merger is (17 + 25 - 9)/53 because 9 stockholders who are in the 200 to under 1000 shares category are also opposed. This gives us 33/53 or approximately 62.3%.
• d. The probability that a stockholder holds under 200 shares given that he/she is opposed to the merger is 5/25 or 20.0%.

Answer 2

The probabilities calculated for the respective parts of the question are: (a) 35.8% for a stockholder holding 1000 or more shares, (b) 17.0% for a stockholder holding less than 200 shares and in favor of the merger, (c) 62.3% for holding between 200 to under 1000 shares or being opposed to the merger, and (d) 20.0% for holding under 200 shares given that the stockholder is opposed.

Step-by-step explanation:

The given problem pertains to the field of probability and involves calculating various probabilities concerning stockholders' opinions and the number of shares they hold in a corporation.

a. To find the probability that a stockholder holds 1000 or more shares, we divide the number of stockholders with 1000 or more shares by the total number of stockholders:

Probability (%) = (Number of stockholders with 1000 or more shares / Total number of stockholders) × 100 = (19 / 53) × 100 ≈ 35.8%

b. To find the probability that a stockholder holds less than 200 shares and is in favor of the merger, we look at the joint occurrence for both conditions:

Probability (%) = (Number of stockholders under 200 shares and in favor / Total number of stockholders) × 100 = (9 / 53) × 100 ≈ 17.0%

c. The probability that a stockholder holds between 200 to under 1000 shares or is opposed to the merger involves considering the union of two events:

Probability (%) = ((Number of stockholders with 200 to under 1000 shares) + (Number of stockholders opposed) - (Both conditions overlap) / Total number of stockholders) × 100 = ((17 + 25 - 9) / 53) × 100 ≈ 62.3%

d. Finally, to find the probability that a stockholder holds under 200 shares, given that they are opposed to the merger, we use conditional probability:

Probability (%) = (Number of stockholders under 200 shares and opposed / Number of stockholders opposed) × 100 = (5 / 25) × 100 = 20.0%