Two fair cubes, numbered 1 through 6, are tossed at the same time in order to advance on a game board. What is the probability that a 6 is rolled on both number cubes?
A. [tex]$\frac{1}{30}$[/tex]
B. [tex]$\frac{1}{6}$[/tex]
C. [tex]$\frac{1}{36}$[/tex]
D. [tex]$\frac{1}{3}$[/tex]
Answer (1)
The probability of rolling a 6 on a single die is 6 1 .
Since the two dice are independent, we multiply the probabilities.
Calculate the probability: 6 1 × 6 1 = 36 1 .
The probability of rolling a 6 on both dice is 36 1 .
Explanation
Understand the problem When two fair six-sided dice are tossed, we want to find the probability of rolling a 6 on both dice. Each die has 6 possible outcomes, numbered 1 through 6.
Probability of rolling a 6 on one die The probability of rolling a 6 on a single die is 6 1 , since there is one favorable outcome (rolling a 6) out of six possible outcomes (1, 2, 3, 4, 5, 6).
Probability of rolling a 6 on both dice Since the two dice are independent, the probability of rolling a 6 on both dice is the product of the probabilities of rolling a 6 on each die. This is because the outcome of one die does not affect the outcome of the other die.
Calculate the final probability To calculate the probability of rolling a 6 on both dice, we multiply the probability of rolling a 6 on the first die by the probability of rolling a 6 on the second die: P ( 6 on both dice ) = P ( 6 on first die ) × P ( 6 on second die ) = 6 1 × 6 1 = 36 1 . Therefore, the probability of rolling a 6 on both dice is 36 1 .
Examples
This type of probability calculation is useful in many games involving dice, such as board games or casino games. For example, if a game requires rolling a specific combination on two dice to win a prize, understanding the probability helps in assessing the odds of winning. In this case, the probability of rolling two 6s is 36 1 , which means that on average, you would expect to roll two 6s once every 36 attempts.
