A rare species of bird visits a sanctuary, with an average rate of 1 bird every 2 days. Given this, what's the probability that:
At least 3 birds will visit the sanctuary in the next 3 days? （Choose the closest answer）

Question

BlackTom AI · Accepted Answer

First, I translate the problem into a standard probability model: events occur at an average rate of 1 bird every 2 days, so over 3 days the expected number (lambda) is 3*(0.5) = 1.5. We are asked for the probability that at least 3 birds arrive in that 3-day interval, which is P(X ≥ 3) for X ~ Poisson(λ = 1.5).

Option 1: 0.013. This is far too small for P(X ≥ 3) with λ = 1.5. To see why, note that the Poisson probabilities for 0, 1, or 2 events with λ = 1.5 already sum to well over 0.8, making P(X ≥ 3) roughly 0.19, so 0.013 underestimates the tail by a large margin.

Option 2: 0.191. This aligns with the actual Poisson tail calculation. Compute e^{-1.5} ≈ 0.22313, then P(0) = 0.22313, P(1) = 1.5×0.22313 ≈ 0.33470, P(2) = (1.5^2/2)×0.22313 ≈ 0.25036. Summing P(0)–P(2) gives ≈ 0.80819, so P(X ≥ 3) ≈ 1 − 0.80819 ≈ 0.19181, which matches this option closely.

Option 3: 0.224. This value is slightly larger than the computed 0.1918, which would correspond to a scenario with a slightly higher lambda or a rougher approximation. Since the exact Poisson tail for λ = 1.5 is around 0.1918, 0.224 overstates the probability.

Option 4: 0.414. This is substantially higher than the Poisson tail for λ = 1.5. Getting as high as 0.414 would require a much larger expected count or a misinterpretation of the event (for example, counting something other than X ≥ 3 in a 3-day window). In the Poisson model with the given rate, this is not realistic.

Overall, the calculation shows the sum of probabilities for 0, 1, and 2 events is about 0.808, leaving a tail probability around 0.192 for 3 or more birds. The closest choice among the options is the one near 0.191. The other options diverge from the Poisson result by either being far too small or too large, reflecting common errors such as undercounting the probability mass below 3 or approximating the tail poorly.

MGS3100W02 Week 7 quiz

A rare species of bird visits a sanctuary, with an average rate of 1 bird every 2 days. Given this, what's the probability that: At least 3 birds will visit the sanctuary in the next 3 days? （Choose the closest answer）

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MGS*3100*W02 Week 7 quiz