Let's answer each question one by one.
Question 1
In order to find the probability of Outcome Y given that Outcome 2 occurs, we need to look at the events defined in the table. The denominator of the fraction we use in the formula P(A|B) = n(A and B) / n(B) is the total number of occurrences of Outcome 2.
From the table:
- Outcome 2 occurs 9 times for Outcome X and 4 times for Outcome Y.
- Therefore, the total occurrences for Outcome 2 is 9 + 4 = 13.
The answer is: 13
Question 2
The problem asks for the probability that their second child is a girl, given that their first child is a boy. In a tree diagram illustrating the combinations of children's genders, the possible scenarios given that the first child is a boy (B) are:
- B (first), B (second)
- B (first), G (second)
Out of these scenarios, only one scenario (B, G) has a girl as the second child. Therefore, the probability that the second child is a girl (given the first child is a boy) is:
P(G | B) = Number of favorable outcomes for G / Total outcomes given first child is B = 1 / 2.
The answer is: 1/2
Question 3
Based on the Venn diagram where:
- Event A (sum is odd): Odd sums from rolling two dice (1+1, 1+3, etc.)
- Event B (sum is 6 or greater): Possible sums from 6 to 12.
We need to find the probability that a sum is odd given that the sum is 6 or greater (P(A | B)).
The odd sums that are 6 or greater from sums (7, 9, 11) occur. The odd sums in our region of interest (sums 6 or greater) are 7 and 9.
- Total sums that are 6 or greater: 6, 7, 8, 9, 10, 11, 12 gives us 7 possible outcomes: {6, 7, 8, 9, 10, 11, 12}
- Favorable outcomes from this set that are odd (sums {7, 9, 11}):
Thus, the probability P(A|B) = (Number of favorable outcomes for A) / (Total outcomes for B) = 3 / 7.
The answer is: 3/7
Question 4
The correct way to determine if the outcome of two events A and B is independent is:
Show that P(A and B) = P(A) * P(B).
This means that the occurrence of one event does not affect the occurrence of the other event.
The answer is: Show that P(A and B) = P(A) * P(B).
Question 5
To determine if the two events A (shape is a triangle) and B (shape is yellow) are independent, we can check the formula for independence:
P(A and B) = P(A) * P(B).
- Total triangles = 50, yellow triangles = 30 (P(A) = 50/80; P(B) = 48/80)
- Calculate P(A and B) = P(yellow triangle) = Number of yellow triangles out of total shapes = 30/80.
Now check:
- P(A) * P(B) = (50/80) * (48/80) = 2400/6400 = 3/8 (not equal to 30/80)
Since P(Yellow Triangle) is not equal to P(A) * P(B), the events are not independent.
The answer is: No, because P(A and B) ≠ P(A) * P(B).