One of two cards is black on one side and white on the other side. The second card is black on both sides. One card is selected at random and the side facing up is black. What is the probability that the other side of the card is white?
There are eight possible card arrangements with equal probability. The first of each pair listed below is the first draw. B' is one of the black sides of the two-sided black card.
The possibilities are:
B/W and B/B'; B/W and B'/B; W/B and B/B'; W/B and B/B', B/B' and W/B; B/B' and B/W; B'/B and W/B; B'/B and B/W. Six of these possibilities give Black in the first draw. The only possibilities after getting a B or B' are the following six:
B/W and B/B'; B/W and B'/B; B/B' and W/B; B/B' and B/W; B'/B and W/B; B'/B and B/W.
Two of those six result in white for the second draw. The probability of white is 1/3.
To solve this problem, we can use conditional probability.
Let's analyze the possibilities:
1. Selecting the card with one black side and one white side:
- Probability of selecting this card = 1/2
- Probability of showing the black side = 1/2
- Probability of the other side being white = 1/2
2. Selecting the card with both sides black:
- Probability of selecting this card = 1/2
- Probability of showing the black side = 1
- Probability of the other side being white = 0
Since we know that the side facing up is black, we only need to consider the cases where the selected card has a black side:
- In 1 out of 2 cases, we have the card with one black side and one white side.
- In the remaining 1 out of 2 cases, we have the card with both sides black.
Therefore, the probability that the other side of the card is white, given that the chosen side is black, is calculated using Bayes' theorem:
P(white side | black side) = (P(white side) * P(black side | white side)) / P(black side)
P(white side) = 1/2 (chance of selecting the card with one black and one white side)
P(black side | white side) = 1/2 (chance of black side facing up if the card has one black and one white side)
P(black side) = 1/2 (chance of selecting a card with a black side)
Therefore,
P(white side | black side) = (1/2 * 1/2) / (1/2)
= 1/2
Hence, the probability that the other side of the card is white, given that the chosen side is black, is 1/2 or 50%.
To determine the probability, let's first consider all the possible outcomes and then figure out the favorable outcomes.
There are a total of 3 possible outcomes:
1) Card 1 is selected, and the black side is facing up.
2) Card 1 is selected, and the white side is facing up.
3) Card 2 is selected, and one of the black sides is facing up.
Out of these 3 possible outcomes, only 2 are favorable to the situation in which the other side of the selected card is white:
1) Card 1 is selected, and the white side is facing up.
2) Card 2 is selected, and the other side (white side) is facing up.
Since we know that the side facing up is black, we can exclude outcome 2 from the total outcomes because it is not possible. Therefore, we are left with 2 total outcomes:
1) Card 1 is selected, and the black side is facing up.
2) Card 2 is selected, and one of the black sides is facing up.
Out of these 2 possible outcomes, only 1 is favorable, which is when Card 1 is selected, and the white side is facing up.
Hence, the probability that the other side of the card is white is 1 out of 2, or in decimal form, 0.5, or 50%.