a. Determine the probability that at most one 2 is rolled. b. Determine the probability that a sum of 4 is not rolled or the red die shows a 2. c. Determine the probability that th... a. Determine the probability that at most one 2 is rolled. b. Determine the probability that a sum of 4 is not rolled or the red die shows a 2. c. Determine the probability that the sum of 5 is rolled and the blue die is showing a 1 or a 3.
Understand the Problem
The question asks us to calculate three probabilities based on rolling a four-sided blue die and a three-sided red die. We need to consider the possible outcomes of each roll and then calculate the probabilities of specific events involving the numbers rolled on each die in three subquestions.
Answer
$P(\text{Blue} > \text{Red}) = \frac{1}{2}$ $P(\text{Blue} = \text{Red}) = \frac{1}{4}$ $P(\text{Blue} + \text{Red} = 5) = \frac{1}{4}$
Answer for screen readers
$P(\text{Blue} > \text{Red}) = \frac{1}{2}$
$P(\text{Blue} = \text{Red}) = \frac{1}{4}$
$P(\text{Blue} + \text{Red} = 5) = \frac{1}{4}$
Steps to Solve
- Determine the sample space
Since the blue die has 4 sides and the red die has 3 sides, there are $4 \times 3 = 12$ possible outcomes. We represent each outcome as an ordered pair $(b, r)$, where $b$ is the result of the blue die and $r$ is the result of the red die.
The sample space $S$ is: $S = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (4, 3)}$
- Calculate Probability of Blue > Red: P(Blue > Red)
We need to count the outcomes where the number on the blue die is greater than the number on the red die. These outcomes are: $(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)$. There are 6 such outcomes. So, $P(\text{Blue} > \text{Red}) = \frac{6}{12} = \frac{1}{2}$.
- Calculate Probability of Blue = Red: P(Blue = Red)
We need to count the outcomes where the number on the blue die is equal to the number on the red die. These outcomes are: $(1, 1), (2, 2), (3, 3)$. There are 3 such outcomes. So, $P(\text{Blue} = \text{Red}) = \frac{3}{12} = \frac{1}{4}$.
- Calculate Probability of Blue + Red = 5: P(Blue + Red = 5)
We need to count the outcomes where the sum of the numbers on the blue and red dice is equal to 5. These outcomes are: $(2, 3), (3, 2), (4, 1)$. There are 3 such outcomes. So, $P(\text{Blue} + \text{Red} = 5) = \frac{3}{12} = \frac{1}{4}$.
$P(\text{Blue} > \text{Red}) = \frac{1}{2}$
$P(\text{Blue} = \text{Red}) = \frac{1}{4}$
$P(\text{Blue} + \text{Red} = 5) = \frac{1}{4}$
More Information
The probability calculations are based on the assumption that both dice are fair, meaning each outcome on each die is equally likely.
Tips
A common mistake is not listing out all possible outcomes in the sample space or miscounting the favorable outcomes for each event. To avoid this, it's helpful to systematically list all possibilities.
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