You have 5 cards, 4 are red and one is black. All cards are face-down and not shuffled. Your goal is to pick a black card. In this case, you either pick a black card or not. The result is not revealed. At least one of the remaining cards is black, and this is turned over. The last card will either be red or not. We want to know what the probability of picking a red card.
This problem relates to the topic on the Monty Hall Problem.
By assumptions, the switching strategy has a 4/5 probability of picking a red card, while keeping the initial choice has a 1/5 probability.
When the player first makes their choice, there is a 4/5 chance that the red card is behind one of the cards not chosen (still face-down). This probability does not change after the dealer reveals a black card behind one of the unchosen cards. When the dealer provides information about the four unchosen (revealing that one of them does not have the red card behind it), the 4/5 chance of the red card being behind one of the unchosen cards rests on the unchosen and unrevealed card, as opposed to the 1/5 chance of the red card being behind the card the player chose initially.
The given probabilities depend on specific assumptions about how the dealer and player choose their cards. An important insight is that, with these standard conditions, there is more information about cards 2, 3, 4, and 5 than was available at the beginning of the game when card 1 was chosen by the player: the dealer's action adds value to the card not eliminated, but not to the one chosen by the player originally. Another insight is that switching cards is a different action from choosing between the four remaining cards at random, as the former action uses the previous information and the latter does not. Other possible behaviors of the dealer than the one described can reveal different additional information, or none at all, leading to different probabilities.
Card 1 = {Black, Black, Black, Black, Red}
Card 2 = {Black, Black, Black, Red, Black}
Card 3 = {Black, Black, Red, Black, Black}
Card 4 = {Black, Red, Black, Black, Black}
Card 5 = {Red, Black, Black, Black, Black}
A player who stays with the initial choice wins in only one out of five of these equally likely possibilities, while a player who switches wins in four out of five.
The conditional probability of winning by switching given the player initially picks card 1 and the dealer flips over card 3 is the probability for the event "red card is behind card 2 and dealer flips over card 3" divided by the probability for "dealer flips over card 3". These probabilities can be determined referring to the conditional probability table below, or to an equivalent decision tree. The conditional probability of winning by switching is 1/5 / [1/5 + (1/5)(1/4)] = 1/5 / [1/5 + 1/20] = 1/5 / 1/4 = 1/5*(4/1) = 4/5 or 0.8.
This problem will have various answer depending on how the player picks its card and how the dealer handles the remaining cards.