An official from the Australian Securities and Investments Commission estimates that 75% of all investment bankers have profited from the use of insider information. If 15 investment bankers are selected at random from the Commission’s registry, find the probability that all 15 have profited from insider information (4 DP).

Question

BlackTom AI · Accepted Answer

We start by identifying the given data: 75% of investment bankers have profited from insider information, so the probability p that a randomly chosen banker has profited is p = 0.75. We are selecting 15 bankers independently, and we want the probability that all 15 have profited.

Key assumption: independence of each banker’s status with respect to insider information. Under independence, the probability that all 15 have profited is the product of the individual probabilities for each banker, which is p^n with n = 15: (0.75)^{15}.

Compute (0.75)^{15} step by step to verify:
- (0.75)^2 = 0.5625
- (0.75)^4 = (0.5625)^2 ≈ 0.31640625
- (0.75)^5 = (0.75)^4 × 0.75 ≈ 0.2373046875
- (0.75)^10 = [(0.75)^5]^2 ≈ (0.2373046875)^2 ≈ 0.0563135
- (0.75)^{15} = (0.75)^{10} × (0.75)^5 ≈ 0.0563135 × 0.2373046875 ≈ 0.01336

Rounding to four decimal places, (0.75)^{15} ≈ 0.0134.

Notes and potential pitfalls:
- If one were to consider any dependence between bankers, the calculation would be different, but the standard approach for such problems assumes independence unless specified otherwise.
- Ensure the rounding is consistent with the requested precision; here you’ve got 0.0134 when rounded to four decimal places.
- If the problem intended a different interpretation (for example, selecting without replacement from a finite registry with a fixed number who profited), the calculation would require more data about the total population and the number who profited; with the information given, independence and p^n is the typical model.

COMM111_S2 Quiz 3: Random Variables and Discrete Distributions

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