4. A and B are independent with P(A U B) =5/8 and P(A and B') = 7/ 24 Calculate (a) P(B) (b) P(A and B) (c) P(A).
To solve this problem, we can use the formula:
P(A U B) = P(A) + P(B) - P(A and B)
Given that P(A U B) = 5/8, we can plug in the values and solve for P(A) and P(B):
5/8 = P(A) + P(B) - P(A and B)
Since A and B are independent, P(A and B) = P(A) * P(B')
Given that P(A and B') = 7/24, we can replace P(A and B) in the equation above:
5/8 = P(A) + P(B) - (P(A) * P(B'))
Now we can use the equation (P(A and B') = 7/24) to solve for P(A) or P(B'). Rearranging the equation:
P(A) * P(B') = 7/24
Since A and B are independent, P(B') = 1 - P(B), so we have:
P(A) * (1 - P(B)) = 7/24
From this equation, we can solve for P(A), and then use it to find P(B):
P(A) = 7/24 / (1 - P(B))
Now we can substitute the value of P(A) back into the equation 5/8 = P(A) + P(B) - (P(A) * P(B')) to find P(B):
5/8 = (7/24 / (1 - P(B))) + P(B) - (7/24 * P(B))
Multiplying through by (1 - P(B)):
(5/8)(1 - P(B)) = 7/24 + (1 - P(B))(24/24) - (7/24)(1 - P(B))
Simplifying:
5/8 - (5/8)P(B) = 7/24 + 1 - P(B) - (7/24) + (7/24)P(B)
Rearranging and combining like terms:
(5/8)P(B) + P(B) - (7/24)P(B) = 7/24 + 1 - (7/24) - 5/8
Multiplying through by 24 to eliminate fractions:
(15/2)P(B) + 24P(B) - 7P(B) = 7 + 24 - 7 - 15
Simplifying:
31/2 P(B) = 9
Dividing by (31/2):
P(B) = 9 / (31/2)
P(B) = (9 * 2) / 31
P(B) = 18/31
Therefore, (a) P(B) = 18/31.
Now that we know P(B), we can substitute it back into the equation 5/8 = P(A) + P(B) - (P(A) * P(B')) to find P(A):
5/8 = P(A) + 18/31 - (P(A) * (1 - 18/31))
Multiplying through by (1 - 18/31):
(5/8)(1 - 18/31) = 18/31 + (1 - 18/31)P(A) - (18/31)(1 - 18/31)
Simplifying:
(5/8)(13/31) = 18/31 + P(A) - (324/31)
Rearranging and combining like terms:
(5/8)P(A) + P(A) - (324/31) = 18/31 + (405/31)
Multiplying through by 31 to eliminate fractions:
(403/8)P(A) + 31P(A) - 324 = 18 + 405
Simplifying:
(697/8)P(A) + 31P(A) = 423
Adding the fractions:
(697/8)P(A) + (248/8)P(A) = 423
(945/8)P(A) = 423
Dividing by (945/8):
P(A) = 423 / (945/8)
P(A) = (423 * 8) / 945
P(A) = 3384/945
Therefore, (c) P(A) = 3384/945.
Now that we have P(A) and P(B), we can calculate P(A and B):
P(A and B) = P(A) * P(B')
Since A and B are independent, P(B') = 1 - P(B), so we have:
P(A and B) = P(A) * (1 - P(B))
Plugging in the values:
P(A and B) = (3384/945)(1 - 18/31)
P(A and B) = (3384/945)(13/31)
Multiplying fractions:
P(A and B) = (3384 * 13) / (945 * 31)
P(A and B) = 43992 / 29355
Therefore, (b) P(A and B) = 43992/29355.
To calculate the probabilities, we can use the following formulas:
(a) P(B) = P(B and A') + P(B and A) (since A and B are independent)
(b) P(A and B) = P(A) * P(B) (since A and B are independent)
(c) P(A) = P(A and B') + P(A and B) (by the law of total probability)
Given information:
P(A U B) = 5/8
P(A and B') = 7/ 24
Step 1: Calculate P(A and B)
P(A U B) = P(A) + P(B) - P(A and B)
5/8 = P(A) + P(B) - P(A and B)
We can rearrange this equation to find P(A and B) in terms of P(A) and P(B):
P(A and B) = P(A) + P(B) - 5/8
Step 2: Calculate P(A)
P(A) = P(A and B') + P(A and B)
P(A) = 7/24 + P(A) + P(B) - 5/8
We can rearrange this equation to find P(A) in terms of P(B):
P(A) - P(A) = 7/24 + P(B) - 5/8
0 = 7/24 + P(B) - 5/8
Step 3: Calculate P(B)
0 = 7/24 + P(B) - 5/8
P(B) = 5/8 - 7/24
P(B) = 15/24 - 7/24
P(B) = 8/24
P(B) = 1/3
(a) P(B) = 1/3
Step 4: Substitute P(B) back into the equation for P(A) to find the value of P(A)
0 = 7/24 + P(1/3) - 5/8
0 = 7/24 + 8/24 - 5/8
0 = 15/24 - 15/24
0 = 0
Since the equation simplifies to 0 = 0, it means that any value of P(A) would satisfy the equation. Therefore, there isn't a unique value for P(A).
(b) P(A and B) = P(A) * P(B)
Since we don't have a specific value for P(A), we can't calculate P(A and B).
(c) P(A) = Any value
So, P(A) can take on any value.