Get detailed and accurate answers to your questions on IDNLearn.com. Our platform offers reliable and comprehensive answers to help you make informed decisions quickly and easily.
Sagot :
To solve this problem, we need to calculate the probability that both specified events will happen when two six-sided dice are tossed.
### Step 1: Calculate the Probability of Event A
Event A is that the first die does NOT land on 5.
- A six-sided die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
- The favorable outcomes for Event A are: {1, 2, 3, 4, 6}, which are 5 out of the 6 possible outcomes.
Hence, the probability of Event A ([tex]\(P(A)\)[/tex]) is:
[tex]\[ P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total number of possible outcomes}} = \frac{5}{6} \][/tex]
### Step 2: Calculate the Probability of Event B
Event B is that the second die lands on 4.
- A six-sided die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
- The favorable outcome for Event B is: {4}, which is 1 out of the 6 possible outcomes.
Hence, the probability of Event B ([tex]\(P(B)\)[/tex]) is:
[tex]\[ P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total number of possible outcomes}} = \frac{1}{6} \][/tex]
### Step 3: Calculate the Probability of Both Events Occurring
Since Event A and Event B are independent events, the probability that both A and B will occur is given by the product of their individual probabilities.
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Substituting the values of [tex]\(P(A)\)[/tex] and [tex]\(P(B)\)[/tex]:
[tex]\[ P(A \text{ and } B) = \frac{5}{6} \cdot \frac{1}{6} \][/tex]
To multiply these fractions:
[tex]\[ P(A \text{ and } B) = \frac{5 \times 1}{6 \times 6} = \frac{5}{36} \][/tex]
Thus, the probability that both the first die does not land on 5 and the second die lands on 4 is [tex]\( \frac{5}{36} \)[/tex].
### Step 1: Calculate the Probability of Event A
Event A is that the first die does NOT land on 5.
- A six-sided die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
- The favorable outcomes for Event A are: {1, 2, 3, 4, 6}, which are 5 out of the 6 possible outcomes.
Hence, the probability of Event A ([tex]\(P(A)\)[/tex]) is:
[tex]\[ P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total number of possible outcomes}} = \frac{5}{6} \][/tex]
### Step 2: Calculate the Probability of Event B
Event B is that the second die lands on 4.
- A six-sided die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
- The favorable outcome for Event B is: {4}, which is 1 out of the 6 possible outcomes.
Hence, the probability of Event B ([tex]\(P(B)\)[/tex]) is:
[tex]\[ P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total number of possible outcomes}} = \frac{1}{6} \][/tex]
### Step 3: Calculate the Probability of Both Events Occurring
Since Event A and Event B are independent events, the probability that both A and B will occur is given by the product of their individual probabilities.
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Substituting the values of [tex]\(P(A)\)[/tex] and [tex]\(P(B)\)[/tex]:
[tex]\[ P(A \text{ and } B) = \frac{5}{6} \cdot \frac{1}{6} \][/tex]
To multiply these fractions:
[tex]\[ P(A \text{ and } B) = \frac{5 \times 1}{6 \times 6} = \frac{5}{36} \][/tex]
Thus, the probability that both the first die does not land on 5 and the second die lands on 4 is [tex]\( \frac{5}{36} \)[/tex].
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.