Respuesta :
Answer:
p ( A ) = 7/36
p ( B ) = 1/2
p ( A ∩ B ) = 1/12
p ( A / B ) = 1/6
p ( B / A ) = 3/7
Step-by-step explanation:
Given:
- Two dices are rolled together n = 10,000 of times
Find:
a) Identify which rolls of the dice are in the event A, the dice add up to a perfect square (4 or 9).
Determine what proportion of the 10,000 rolls are in A.
Solution:
- Identity all outcomes per roll such that the sum is a perfect square i.e 4 or 9.
Sum is 4 or 9: ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 1 ) , ( 3 , 6 ) , ( 5 , 4 ) , ( 4 , 5 ) , ( 6 , 3 )
Possible outcomes per roll = 7.
Total Outcomes of sums per roll = 36
- Since each roll is independent from each other the proportion of 10,000 rolls in event A would be:
Event ( A ) = n* ( Possible Outcomes / Total Outcomes )
Event ( A ) = 10,000* ( 7 / 36 ) = 1944.44
Proportion p(A) = Event (A) / n = 1944.44 / 10,000 = 0.1944
Find:
b) Identify which rolls of the dice are in the event B, the dice add up to an even number.
Determine what proportion of the 10,000 rolls are in B.
Solution:
- Identity all outcomes per roll such that the sum is even 2 , 4 , 6 , 8 , 10 , 12.
Sum is even: ( 1 , 1 ) , ( 1 , 3 ) , ( 1 , 5 ) , ( 2 , 2 ) , ( 2 , 4 ) , ( 2 , 6 ) , ( 3 , 1 ) , ( 3 , 3 ) , ( 3 , 5 ) , ( 4 , 2 ) , ( 4 , 4 ) , ( 4 , 6 ) ( 5 , 1 ) , ( 5 , 3 ) , ( 5 , 5 ) , ( 6 , 2 ) , ( 6 , 4 ) , ( 6 , 6 )
Possible outcomes per roll = 18.
Total Outcomes of sums per roll = 36
- Since each roll is independent from each other the proportion of 10,000 rolls in event B would be:
Event ( B ) = n* ( Possible Outcomes / Total Outcomes )
Event ( B ) = 10,000* ( 18 / 36 ) = 5,000
Proportion p(B) = Event (B) / n = 5,000 / 10,000 = 0.5
Find:
c) Find out which rolls are in A ∩ B. Find the proportion that are in A ∩ B. How does that compare to the proportion in A multiplied by the proportion that are in B?
- The events outcomes common in both events are as follows:
A ∩ B = ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 1 )
Possible outcomes per roll = 3.
Total Outcomes of sums per roll = 36
- Since each roll is independent from each other the proportion of 10,000 rolls in event A ∩ B would be:
Event ( A ∩ B ) = n* ( Possible Outcomes / Total Outcomes )
Event ( A ∩ B ) = 10,000* ( 3 / 36 ) = 833.33
Proportion p(A ∩ B) = Event (A ∩ B) / n = 833.333 / 10,000 = 1/12
p(A) * p(B) = 0.5*0.1944 = 0.09722 > 1/12
Find:
d) Of the rolls in which B occurs what proportion of those rolls are also in A. How does this compare to the proportion you computed in part (b)?
- Possible outcomes common in both A and B = 3
- p ( A / B ) = p ( A ∩ B ) / p(B)
= ( 1 / 12 ) / 0.5
= 1 / 6
- p ( B / A ) = p ( A ∩ B ) / p(A)
= ( 1 / 12 ) / (7/36)
= 3 / 7
