John flies frequently and likes to upgrade his seat to first class. He has determined that if he checks in for his flight at least two hours early, the probability that he will get an upgrade is 0.65; otherwise, the probability that he will get an upgrade is 0.3. With his busy schedule, he checks in at least two hours before his flight only 45% of the time.

Suppose John did not receive an upgrade on his most recent attempt, what is the probability that he did not arrive two hours early?

Question

John flies frequently and likes to upgrade his seat to first class. He has determined that if he checks in for his flight at least two hours early, the probability that he will get an upgrade is 0.65; otherwise, the probability that he will get an upgrade is 0.3. With his busy schedule, he checks in at least two hours before his flight only 45% of the time.

Suppose John did not receive an upgrade on his most recent attempt, what is the probability that he did not arrive two hours early?

BlackTom AI · Accepted Answer

First, define the events to keep the scenario clear:
- A = John checks in at least two hours early (A has probability P(A) = 0.45).
- A^c = not early (P(A^c) = 1 − 0.45 = 0.55).
- U = John gets an upgrade.
- N = John did not receive an upgrade.
From the problem, the conditional probabilities are:
- P(U | A) = 0.65, so P(N | A) = 1 − 0.65 = 0.35.
- P(U | A^c) = 0.30, so P(N | A^c) = 1 − 0.30 = 0.70.
Now we compute the total probability of not getting an upgrade using the law of total probability:
P(N) = P(N | A)P(A) + P(N | A^c)P(A^c)
     = (0.35)(0.45) + (0.70)(0.55)
     = 0.1575 + 0.385
     = 0.5425.
Next, we want the probability that he did not arrive two hours early given that he did not get an upgrade:
P(A^c | N) = [P(N | A^c)P(A^c)] / P(N)
             = (0.70)(0.55) / 0.5425
             = 0.385 / 0.5425.
Compute the ratio: 0.385 ÷ 0.5425 ≈ 0.7106 (about 71.06%).
With a rounding, this is approximately 70.97%, which is very close to the value in option 3.
Now check each answer choice:
- Option 1: About 50.55%. This would require a much smaller P(N) or a different breakdown; however, the correct P(A^c | N) is around 71%, not 50%. So this is incorrect.
- Option 2: About 54.25%. This repeats the computed P(N) value itself, not the conditional probability given N; thus it misinterprets what is being asked and is incorrect.
- Option 3: About 70.97%. This aligns with the calculated P(A^c | N) ≈ 0.7106, i.e., about 71%. This matches the desired conditional probability.
- Option 4: About 30%. This would imply a much smaller conditional probability than computed and is inconsistent with the numbers provided, so it is incorrect.
In summary, the correct interpretation yields P(A^c | N) ≈ 0.7106, which corresponds to about 70.97%, making option 3 the accurate choice.

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