Graduate Management Admission Test (GMAT) Practice Test

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In calculating the probability of at least one event occurring from two draws, which formula is appropriate?

1 - (1-p)(1-q)
p + q
(1-p) + (1-q)
pq

The correct answer is: 1 - (1-p)(1-q)

The formula $1 - (1-p)(1-q)$ is appropriate for calculating the probability of at least one event occurring from two independent draws. To understand why this is the correct choice, it's useful to analyze what $p$ and $q$ represent in this context. Here, $p$ is the probability of the first event occurring, while $q$ is the probability of the second event occurring. The expression $1-p$ gives the probability that the first event does not occur, and $1-q$ gives the probability that the second event does not occur. The product $(1-p)(1-q)$ therefore represents the probability that neither event occurs in the two draws. By subtracting this product from 1, we obtain the probability that at least one of the events occurs. This approach effectively captures the complementary relationship of the scenario, where we first calculate the situation we want to avoid (neither event occurring) and subtract that from the total probability, which is 1. The other formulas provided do not appropriately capture the scenario described. Adding $p$ and $q$ would miscalculate the likelihood by assuming that these events are mutually exclusive, which is not necessarily true.