Two fair dice, one blue and one red, are tossed, and the up face
on each die is recorded. Define the following events:
E : { The sum of the numbers is even } F : { The numbers are equal } Find the following probabilities:
(a) P(E) =
(b) P(F) =
(c) P(E|F) =
Are events E and F independent?
To find the probabilities, we need to determine the number of favorable outcomes over the total number of possible outcomes.
Let's list out all the possible outcomes:
1. Blue die: 1, Red die: 1
2. Blue die: 1, Red die: 2
3. Blue die: 1, Red die: 3
4. Blue die: 1, Red die: 4
5. Blue die: 1, Red die: 5
6. Blue die: 1, Red die: 6
7. Blue die: 2, Red die: 1
8. Blue die: 2, Red die: 2
9. Blue die: 2, Red die: 3
10. Blue die: 2, Red die: 4
11. Blue die: 2, Red die: 5
12. Blue die: 2, Red die: 6
13. Blue die: 3, Red die: 1
14. Blue die: 3, Red die: 2
15. Blue die: 3, Red die: 3
16. Blue die: 3, Red die: 4
17. Blue die: 3, Red die: 5
18. Blue die: 3, Red die: 6
19. Blue die: 4, Red die: 1
20. Blue die: 4, Red die: 2
21. Blue die: 4, Red die: 3
22. Blue die: 4, Red die: 4
23. Blue die: 4, Red die: 5
24. Blue die: 4, Red die: 6
25. Blue die: 5, Red die: 1
26. Blue die: 5, Red die: 2
27. Blue die: 5, Red die: 3
28. Blue die: 5, Red die: 4
29. Blue die: 5, Red die: 5
30. Blue die: 5, Red die: 6
31. Blue die: 6, Red die: 1
32. Blue die: 6, Red die: 2
33. Blue die: 6, Red die: 3
34. Blue die: 6, Red die: 4
35. Blue die: 6, Red die: 5
36. Blue die: 6, Red die: 6
(a) Probability of event E: { The sum of the numbers is even }
The favorable outcomes for event E are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35 (18 outcomes)
The total number of possible outcomes is 36.
P(E) = Total favorable outcomes / Total possible outcomes = 18 / 36 = 1/2
(b) Probability of event F: { The numbers are equal }
The favorable outcomes for event F are: 1, 2, 3, 4, 5, 6 (6 outcomes)
The total number of possible outcomes is 36.
P(F) = Total favorable outcomes / Total possible outcomes = 6 / 36 = 1/6
(c) Probability of event E given event F: P(E|F)
The favorable outcomes for event E and F are: 1, 3, 5 (3 outcomes)
The total number of favorable outcomes for event F is 6.
P(E|F) = Total favorable outcomes / Total favorable outcomes for F = 3 / 6 = 1/2
To check if events E and F are independent, we need to compare P(E) * P(F) with P(E ∩ F).
P(E) * P(F) = (1/2) * (1/6) = 1/12
P(E ∩ F) = 3 / 36 = 1/12
Since P(E) * P(F) = P(E ∩ F) = 1/12, the events E and F are independent.
To find the probabilities of the events E and F, we first need to determine the sample space for this experiment. Since we are tossing two dice, each with 6 faces numbered 1 to 6, the sample space consists of all possible pairs of numbers that can result from the dice toss.
Sample Space = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), ..., (6, 5), (6, 6)}
Now, let's find the probabilities of each event:
(a) P(E) = Probability of the sum of the numbers being even:
To find P(E), we need to determine the number of outcomes in the sample space that satisfy event E. There are overall 36 possible outcomes in the sample space, and for the sum to be even, we have the following possibilities: (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). So, there are 18 outcomes in the sample space that satisfy event E.
P(E) = Number of outcomes satisfying event E / Total number of outcomes in sample space = 18 / 36 = 1/2
(b) P(F) = Probability of the numbers being equal:
To find P(F), we need to determine the number of outcomes in the sample space that satisfy event F. The outcomes in the sample space where the numbers are equal are: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). So, there are 6 outcomes in the sample space that satisfy event F.
P(F) = Number of outcomes satisfying event F / Total number of outcomes in sample space = 6 / 36 = 1/6
(c) P(E|F) = Probability of event E given that event F has occurred:
To find P(E|F), we need to calculate the probability of event E happening given that event F has occurred, i.e., when the numbers are equal. When the numbers are equal, there are three possibilities: (1, 1), (2, 2), (3, 3), each of which satisfies event E (the sum is even). So, there are 3 outcomes in the sample space that satisfy both event E and event F.
P(E|F) = Number of outcomes satisfying both event E and event F / Number of outcomes satisfying event F = 3 / 6 = 1/2
To determine whether events E and F are independent, we need to check if the occurrence of event F has any influence on the probability of event E. If P(E|F) = P(E), then events E and F are independent.
In this case, P(E|F) = 1/2 and P(E) = 1/2. Since they are equal, events E and F are independent.