Shawn is playing a game with a set of cards. Half of them are blue and the other half are yellow. In the game, the player guesses the color of the top card, looks at the card, and returns the card to the deck. The player continues to do this, shuffling the deck after each guess. Shawn has guessed the first three attempts as "blue” and has been correct on each guess. He says he will guess "yellow” for the next card since a yellow card is due to happen. Is Shawn’s reasoning correct?
Answer
A
Yes, half of the cards are yellow, so the fourth card should be yellow to compensate for the first three cards being blue.
B
No, the law of large numbers says that the proportion of yellow cards should approach the true probability after many trials.
C
No, all of the cards have been blue so far, so the next one must also be blue.
D
Yes, Shawn has guessed well so far, so it is clear that he cannot miss.
Explanation
Each guess is made after the card is returned and the deck is shuffled, so the previous draws do not change the next draw’s chances. The probability of blue or yellow on the fourth draw is still 1/2; yellow is not “due.” Over many trials, the proportion of yellow results tends to get closer to 1/2, which is the law of large numbers, but it does not force the next result to be yellow.
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