resulted in a 5 is
Probability of Sum Greater Than 9
This problem involves finding the probability of an event occurring, given that another specific event has already occurred. This is known as conditional probability.
Dice Roll Events Defined
We are rolling two dice, one black and one red, simultaneously. Let's define the events involved:
- Event A: The sum of the numbers shown on the two dice is greater than 9.
- Event C: The black die shows the number 5.
We are asked to find the probability of Event A occurring, given that Event C has occurred. This is represented as $P(A|C)$.
Black Die Condition Analysis
The condition given is that the black die resulted in a 5. This means we only need to consider the outcomes where the black die is 5. The possible outcomes are:
| Black Die | Red Die |
|---|---|
| 5 | 1 |
| 5 | 2 |
| 5 | 3 |
| 5 | 4 |
| 5 | 5 |
| 5 | 6 |
There are 6 possible outcomes when the black die shows a 5, as the red die can take any value from 1 to 6. This set of 6 outcomes forms our reduced sample space for this conditional probability problem.
Sum Greater Than 9 Outcomes
Within this reduced sample space (where the black die is 5), we need to identify the outcomes where the sum of the numbers on both dice is greater than 9.
- Outcome (5, 1): Sum = $5 + 1 = 6$. This is not greater than 9.
- Outcome (5, 2): Sum = $5 + 2 = 7$. This is not greater than 9.
- Outcome (5, 3): Sum = $5 + 3 = 8$. This is not greater than 9.
- Outcome (5, 4): Sum = $5 + 4 = 9$. This is not greater than 9.
- Outcome (5, 5): Sum = $5 + 5 = 10$. This is greater than 9.
- Outcome (5, 6): Sum = $5 + 6 = 11$. This is greater than 9.
The outcomes that satisfy the condition (black die is 5) and the event (sum > 9) are (5, 5) and (5, 6). Therefore, there are 2 favorable outcomes.
Calculating Conditional Probability
The conditional probability $P(A|C)$ is calculated by dividing the number of outcomes that satisfy both Event A and Event C by the number of outcomes that satisfy Event C.
$ P(\text{Sum} > 9 \mid \text{Black Die} = 5) = \frac{\text{Number of outcomes where Sum} > 9 \text{ and Black Die} = 5}{\text{Number of outcomes where Black Die} = 5} $
From our analysis:
- The number of outcomes where the sum is greater than 9 and the black die is 5 is 2.
- The total number of outcomes where the black die is 5 is 6.
So, the probability is:
$ P(\text{Sum} > 9 \mid \text{Black Die} = 5) = \frac{2}{6} $
Simplifying this fraction, we get:
$ \frac{2}{6} = \frac{1}{3} $
Therefore, the probability of obtaining a sum greater than 9, given that the black die resulted in a 5, is $1/3$.
Important Questions from Conditional Probability
A and B are two events such that A̅ and B̅ are mutually exclusive. If P(A) = 0.5 and P(B) = 0.6, then what is the value of P(A|B)?
For two dependent events A and B, it is given that P(A) = 0.2 and P(B) = 0.5. If A ⊆ B, then the values of conditional probabilities P(A|B) and P(B|A) are respectively
In a bulb factory, machines P, Q and R manufacture respectively 25%, 35% and 40% of the total. Of their output 5, 4 and 2 percent respectively are defective bulbs. A bulb is drawn at random and it is found to be defective. What is the probability that it was manufactured by machine Q?
Let X1 and X2 be independent random variables each having geometric distribution qk p ; k = 0, 1, 2, …. Then the conditional distribution of X1 given X1 + X2 is
Find the probability that they speak the same fact if A speaks truth in 60% cases and B speaks truth in 75% cases?
