Question

predicting the winner
you must roll a total of 9 on two number cubes to win the game. alex must roll a 6. who is more likely to win the game on the next turn? explain.

Explanation:

Step1: Find total number of outcomes

When rolling two number - cubes, each cube has 6 possible outcomes. So the total number of outcomes when rolling two cubes is \(6\times6 = 36\).

Step2: Find number of ways to get a sum of 9

The pairs of numbers on the two cubes that sum to 9 are \((3,6)\), \((4,5)\), \((5,4)\), \((6,3)\). So there are 4 ways to get a sum of 9. The probability of getting a sum of 9, \(P(9)=\frac{4}{36}=\frac{1}{9}\).

Step3: Find number of ways to get a sum of 6

The pairs of numbers on the two cubes that sum to 6 are \((1,5)\), \((2,4)\), \((3,3)\), \((4,2)\), \((5,1)\). So there are 5 ways to get a sum of 6. The probability of getting a sum of 6, \(P(6)=\frac{5}{36}\).

Step4: Compare probabilities

Since \(\frac{5}{36}>\frac{1}{9}=\frac{4}{36}\), Alex is more likely to win.

Answer:

Alex is more likely to win because the probability of rolling a 6 on two number - cubes (\(\frac{5}{36}\)) is greater than the probability of rolling a 9 on two number - cubes (\(\frac{4}{36}\)).