A bag holds four orbs, two red and two blue, which are drawn out one at a time without replacement. Before each draw you announce a colour and earn $$1$ if the next orb matches it. With the best guessing strategy, what are your expected total winnings?

Guessing colours of two red and two blue orbs

Question

A bag holds four orbs, two red and two blue, which are drawn out one at a time without replacement. Before each draw you announce a colour and earn $\text{\char36}1$ if the next orb matches it. With the best guessing strategy, what are your expected total winnings?

Short answer

$17/6$ dollars. Always guess the colour with more orbs remaining. The first guess is right with probability $\tfrac12$ , the second $\tfrac23$ , the third $\tfrac23$ (certain if the first two matched, a coin flip otherwise), and the last is always right: $\tfrac12+\tfrac23+\tfrac23+1=\tfrac{17}{6}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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