Asked by zoie
A wheel at a carnival game has
sectors of equal size, with
red sectors and
black sectors.
At the end of a spin of the wheel, the pointer is equally likely to land on a red sector or a black sector.
A boy observes four consecutive red spins and guesses the next spin will be red.
When asked why, he says the wheel is on a “hot streak of red.”
Is his reasoning accurate? Why or why not?
All Answers 1
Answered by
GPT-5 mini
AI
No — his reasoning is not accurate if the wheel is fair and spins are independent.
Because each spin is equally likely red or black and independent of previous spins, the probability the next spin is red is 1/2 regardless of the past. Formally,
P(next = red | previous four = red) = P(five reds)/P(four reds) = (1/2^5)/(1/2^4) = 1/2.
Saying the wheel is on a “hot streak” is an example of the hot‑hand/gambler’s‑fallacy thinking that past independent outcomes change future ones. (A caveat: if you did not actually know the wheel was fair or independent, observing four reds could legitimately make you suspect the wheel is biased; in that case a Bayesian update would raise your estimate of the chance of red.)
Because each spin is equally likely red or black and independent of previous spins, the probability the next spin is red is 1/2 regardless of the past. Formally,
P(next = red | previous four = red) = P(five reds)/P(four reds) = (1/2^5)/(1/2^4) = 1/2.
Saying the wheel is on a “hot streak” is an example of the hot‑hand/gambler’s‑fallacy thinking that past independent outcomes change future ones. (A caveat: if you did not actually know the wheel was fair or independent, observing four reds could legitimately make you suspect the wheel is biased; in that case a Bayesian update would raise your estimate of the chance of red.)
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