An event has a probability of .6 of occurring. Given 8 trials, what is the probability it occurs at least 2 times?

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BlackTom AI · Accepted Answer

The problem asks for the probability that the event occurs at least 2 times in 8 trials, where each trial has probability 0.6 of success. This is a binomial probability scenario with n = 8 and p = 0.6, and the target is P(X ≥ 2).

Option 1: '1-Binomial(k=1 n=8, p=0.6, Cumulative=TRUE)'. This reads as 1 minus the cumulative probability up to k = 1, i.e., 1 - P(X ≤ 1). Since P(X ≥ 2) is indeed equal to 1 - P(X ≤ 1) for a binomial distribution, this option correctly expresses the desired calculation in the standard cumulative form.

Option 2: 'Binomial(k=2, n=8, p=0.6, Cumulative=TRUE)'. This phrasing suggests P(X ≤ 2) if the cumulative flag is TRUE, or some nonstandard interpretation of a cumulative binomial with k = 2. However, the target is not the probability of at most 2 successes, but at least 2 successes, so this option does not directly represent P(X ≥ 2).

Option 3: 'Binomial(k=2, n=8, p=0.6, Cumulative=TRUE)'. This is essentially the same wording as Option 2, and shares the same issue: it would correspond to P(X ≤ 2) under a cumulative interpretation, which again does not equal P(X ≥ 2). The intended quantity is the complement of at most 1 success, not at most 2.

Option 4: '1-Binomial(k=6, n=8, p=0.6, Cumulative=TRUE)'. This would compute 1 minus P(X ≤ 6), which equals P(X ≥ 7). That is a probability of 7 or 8 successes, not at least 2 successes, so this option misidentifies the target range entirely.

In summary, the correct approach to the requested probability is to take the complement of the probability of fewer than 2 successes (i.e., 0 or 1 success) and express it as 1 - P(X ≤ 1). The other options either attempt to represent P(X ≤ 2) or P(X ≥ 7), which do not match the required event of X ≥ 2 across 8 trials.

ERMCPS5350_006_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 5. Quiz for Class 5

An event has a probability of .6 of occurring. Given 8 trials, what is the probability it occurs at least 2 times?

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