In the game of craps, the player rolls two balanced dice, 36 equally likely outcomes, compute the probability the player looses on the first roll if the sum of the dice is 3, 4, or 6.
draw a box or matrix with your 36 possibilities
make an x for win, an o for lose(3,4,6)
0 1 2 3 4 5 6
1 x o o x o x
2 o o x o x x
3 o x o x x x
4 x o x x x x
5 o x the rest x
6
number of o s = 10
so 10/36
To compute the probability of losing on the first roll in the game of craps, given that the sum of the two dice is 3, 4, or 6, we need to determine how many outcomes result in a loss and divide that by the total number of equally likely outcomes.
In craps, a player loses on the first roll if the sum of the two dice is 2, 3, or 12. Let's calculate the number of outcomes that result in a loss for each possible sum:
1. Sum = 2: There is only one way to roll a sum of 2, which is by rolling a 1 on both dice. So, there is 1 outcome that results in a loss.
2. Sum = 3: There are two ways to roll a sum of 3: (1, 2) and (2, 1). So, there are 2 outcomes that result in a loss.
3. Sum = 4: There are three ways to roll a sum of 4: (1, 3), (2, 2), and (3, 1). So, there are 3 outcomes that result in a loss.
4. Sum = 5: There are four ways to roll a sum of 5: (1, 4), (2, 3), (3, 2), and (4, 1). However, since we're only considering the sums of 3, 4, and 6, we exclude this sum as it doesn't lead to a loss.
5. Sum = 6: There are five ways to roll a sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1). So, there are 5 outcomes that result in a loss.
Now, let's calculate the total number of equally likely outcomes. Since each dice has 6 sides, there are 6 possible outcomes for each dice, resulting in a total of 6 x 6 = 36 equally likely outcomes.
To find the probability, we divide the number of outcomes resulting in a loss by the total number of equally likely outcomes:
Probability of losing on first roll (sum = 3, 4, or 6) = (Number of outcomes resulting in a loss) / (Total number of equally likely outcomes)
Probability of losing on first roll (sum = 3, 4, or 6) = (1 + 2 + 3) / 36
Probability of losing on first roll (sum = 3, 4, or 6) = 6 / 36
Probability of losing on first roll (sum = 3, 4, or 6) = 1 / 6
Therefore, the probability of losing on the first roll if the sum of the dice is 3, 4, or 6 is 1/6.