Trinity created a game that uses 8 cards numbered 1 to 8 and a six-sided cube numbered 1 to 6. Each player chooses one card and rolls the cube to determine the number of spaces to move on the game board. What is the probability that Trinity will choose a card greater than 6 and roll a 6 on the cube? A. 1/18 B. 1/16 C. 1/24 D. 1/12
The correct answer and explanation is:
Let’s analyze the problem step-by-step:
Problem summary:
- There are 8 cards numbered from 1 to 8.
- There is a six-sided cube (die) numbered from 1 to 6.
- Trinity chooses one card and rolls the die.
- We want the probability that Trinity chooses a card greater than 6 and rolls a 6 on the cube.
Step 1: Understand the event “choosing a card greater than 6”
- Cards are numbered 1 to 8.
- Cards greater than 6 are: 7 and 8.
- So there are 2 favorable cards out of 8 total cards.
Probability of choosing a card > 6 is: P(card>6)=28=14P(\text{card} > 6) = \frac{2}{8} = \frac{1}{4}
Step 2: Understand the event “rolling a 6 on the cube”
- The cube is numbered 1 through 6.
- The probability of rolling a 6 is:
P(roll 6)=16P(\text{roll } 6) = \frac{1}{6}
Step 3: Combine the two independent events
Choosing a card and rolling a die are independent events. So, the probability that both happen together is the product of the individual probabilities: P(card>6 and roll 6)=P(card>6)×P(roll 6)=14×16=124P(\text{card} > 6 \text{ and roll } 6) = P(\text{card} > 6) \times P(\text{roll } 6) = \frac{1}{4} \times \frac{1}{6} = \frac{1}{24}
Final answer:
Probability = 124\boxed{\frac{1}{24}}
Corresponding to option C. 1/24
Explanation in about 300 words:
In this problem, we are calculating the probability of two independent events happening together: selecting a card with a value greater than 6 and rolling a 6 on a six-sided cube. There are 8 cards, numbered from 1 to 8, and only two of them (7 and 8) are greater than 6. Hence, the probability of selecting such a card is 2 out of 8, or simplified, 1/4.
The cube, being a standard six-sided die, has equal probability for each number from 1 to 6. The probability of rolling a 6 is therefore 1/6.
Since the card selection and die roll are independent events — the outcome of one does not affect the other — the combined probability of both occurring is found by multiplying their individual probabilities.
Multiplying 1/4 (choosing a card > 6) by 1/6 (rolling a 6) gives 1/24. This means that out of all possible outcomes (choosing any card and rolling any number), only 1 in 24 will result in the specific scenario where Trinity picks a card greater than 6 and rolls a 6.
This solution assumes each card is equally likely to be chosen and each die face has an equal chance of appearing, which is standard in probability problems unless stated otherwise. Thus, the final probability is 1/24, matching option C.