A safety deposit box contains three ancient Roman coins:
#1 both sides are silver
#2 both sides are gold
#3 one side is silver, the other gold
You are blindfolded, then you pick a coin, then toss it so that it lands on the floor.
At this point, the sample space for the side showing is:
coin # \ side # 1 2
1 silver silver
2 gold gold
3 silver gold
The blindfold comes off, you look down, and you see gold showing.
Given that gold is showing, determine these three probabilities:
P(you drew coin #1)
P(you drew coin #2)
P(you drew coin #3)
To determine the probabilities, we can use conditional probability. The probability of an event A given that event B has occurred is denoted as P(A|B) and is calculated using the formula:
P(A|B) = P(A ∩ B) / P(B)
In this case, we want to determine the probability of drawing each coin, given that gold is showing. Let's calculate the probabilities step by step:
1. P(you drew coin #1):
To calculate the probability of drawing coin #1, given that gold is showing, we need to find the probability of drawing coin #1 and gold showing, divided by the probability of gold showing.
P(you drew coin #1 | gold showing) = P(you drew coin #1 ∩ gold showing) / P(gold showing)
The probability of drawing coin #1 and gold showing is 1/3, as there is only one coin that is gold on both sides. The probability of gold showing is the sum of the probabilities of drawing coin #1 and gold showing, drawing coin #2 and gold showing, and drawing coin #3 and gold showing, which is (1/3 + 1/3 + 1/2) = 5/6.
P(you drew coin #1 | gold showing) = (1/3) / (5/6) = 2/5 = 0.4
So the probability of drawing coin #1, given that gold is showing, is 0.4 or 40%.
2. P(you drew coin #2):
To calculate the probability of drawing coin #2, given that gold is showing, we need to find the probability of drawing coin #2 and gold showing, divided by the probability of gold showing.
P(you drew coin #2 | gold showing) = P(you drew coin #2 ∩ gold showing) / P(gold showing)
The probability of drawing coin #2 and gold showing is 1/3, as there is only one coin that is gold on both sides. The probability of gold showing is the sum of the probabilities of drawing coin #1 and gold showing, drawing coin #2 and gold showing, and drawing coin #3 and gold showing, which is (1/3 + 1/3 + 1/2) = 5/6.
P(you drew coin #2 | gold showing) = (1/3) / (5/6) = 2/5 = 0.4
So the probability of drawing coin #2, given that gold is showing, is 0.4 or 40%.
3. P(you drew coin #3):
To calculate the probability of drawing coin #3, given that gold is showing, we need to find the probability of drawing coin #3 and gold showing, divided by the probability of gold showing.
P(you drew coin #3 | gold showing) = P(you drew coin #3 ∩ gold showing) / P(gold showing)
The probability of drawing coin #3 and gold showing is 1/2, as there are two possible outcomes for coin #3: gold showing or silver showing. The probability of gold showing is the sum of the probabilities of drawing coin #1 and gold showing, drawing coin #2 and gold showing, and drawing coin #3 and gold showing, which is (1/3 + 1/3 + 1/2) = 5/6.
P(you drew coin #3 | gold showing) = (1/2) / (5/6) = 3/5 = 0.6
So the probability of drawing coin #3, given that gold is showing, is 0.6 or 60%.