Answer :
Sure, let's solve the problem step by step. We have a fair dice rolled twice, and we need to find the probabilities for various events defined in terms of heads and tails. To connect the dice rolls to heads and tails, let's define:
- Heads (H): Rolling an even number (2, 4, 6)
- Tails (T): Rolling an odd number (1, 3, 5)
### Total Possible Outcomes
Since the dice is rolled twice:
- Each roll has 6 possible outcomes (1-6)
- Total outcomes for two rolls = 6 6 = 36
### (a) Getting exactly two heads
Two heads implies both numbers rolled are even.
Possible even numbers: 2, 4, 6
Total even numbers = 3
For exactly two heads (both rolls are even):
- Number of favorable outcomes = 3 3 = 9
Probability = Number of favorable outcomes / Total possible outcomes = 9 / 36 = 1 / 4
### (b) No head
No head implies both numbers rolled are odd.
Possible odd numbers: 1, 3, 5
Total odd numbers = 3
For no heads (both rolls are odd):
- Number of favorable outcomes = 3 3 = 9
Probability = Number of favorable outcomes / Total possible outcomes = 9 / 36 = 1 / 4
### (c) Getting at least one head
At least one head means we need to find the complement of the event "no heads" (from part b).
Probability of no head = 1/4
So, probability of at least one head = 1 - probability of no heads = 1 - (1 / 4) = 3 / 4
### (d) Getting a head and a tail
This means one roll is even, and the other is odd. There are two possible combinations: (Head, Tail) or (Tail, Head).
Number of ways to get one even and one odd:
- Even on the first roll and odd on the second roll: 3 3 = 9
- Odd on the first roll and even on the second roll: 3 * 3 = 9
Total number of favorable outcomes: 9 + 9 = 18
Probability = Number of favorable outcomes / Total possible outcomes = 18 / 36 = 1 / 2
### (e) Getting at most one tail
"At most one tail" includes two cases: Zero tails and exactly one tail.
1. Zero tails: Both rolls are even
- We've already calculated this in part (a) as 9 outcomes
2. Exactly one tail:
- This includes (Head, Tail) and (Tail, Head), which we calculated in part (d) as 18 outcomes
Total favorable outcomes for at most one tail = Zero tails + Exactly one tail = 9 + 18 = 27
Probability = Number of favorable outcomes / Total possible outcomes = 27 / 36 = 3 / 4
---
Summary of probabilities:
- (a) Exactly two heads: [tex]\( \frac{1}{4} \)[/tex]
- (b) No head: [tex]\( \frac{1}{4} \)[/tex]
- (c) At least one head: [tex]\( \frac{3}{4} \)[/tex]
- (d) A head and a tail: [tex]\( \frac{1}{2} \)[/tex]
- (e) At most one tail: [tex]\( \frac{3}{4} \)[/tex]
- Heads (H): Rolling an even number (2, 4, 6)
- Tails (T): Rolling an odd number (1, 3, 5)
### Total Possible Outcomes
Since the dice is rolled twice:
- Each roll has 6 possible outcomes (1-6)
- Total outcomes for two rolls = 6 6 = 36
### (a) Getting exactly two heads
Two heads implies both numbers rolled are even.
Possible even numbers: 2, 4, 6
Total even numbers = 3
For exactly two heads (both rolls are even):
- Number of favorable outcomes = 3 3 = 9
Probability = Number of favorable outcomes / Total possible outcomes = 9 / 36 = 1 / 4
### (b) No head
No head implies both numbers rolled are odd.
Possible odd numbers: 1, 3, 5
Total odd numbers = 3
For no heads (both rolls are odd):
- Number of favorable outcomes = 3 3 = 9
Probability = Number of favorable outcomes / Total possible outcomes = 9 / 36 = 1 / 4
### (c) Getting at least one head
At least one head means we need to find the complement of the event "no heads" (from part b).
Probability of no head = 1/4
So, probability of at least one head = 1 - probability of no heads = 1 - (1 / 4) = 3 / 4
### (d) Getting a head and a tail
This means one roll is even, and the other is odd. There are two possible combinations: (Head, Tail) or (Tail, Head).
Number of ways to get one even and one odd:
- Even on the first roll and odd on the second roll: 3 3 = 9
- Odd on the first roll and even on the second roll: 3 * 3 = 9
Total number of favorable outcomes: 9 + 9 = 18
Probability = Number of favorable outcomes / Total possible outcomes = 18 / 36 = 1 / 2
### (e) Getting at most one tail
"At most one tail" includes two cases: Zero tails and exactly one tail.
1. Zero tails: Both rolls are even
- We've already calculated this in part (a) as 9 outcomes
2. Exactly one tail:
- This includes (Head, Tail) and (Tail, Head), which we calculated in part (d) as 18 outcomes
Total favorable outcomes for at most one tail = Zero tails + Exactly one tail = 9 + 18 = 27
Probability = Number of favorable outcomes / Total possible outcomes = 27 / 36 = 3 / 4
---
Summary of probabilities:
- (a) Exactly two heads: [tex]\( \frac{1}{4} \)[/tex]
- (b) No head: [tex]\( \frac{1}{4} \)[/tex]
- (c) At least one head: [tex]\( \frac{3}{4} \)[/tex]
- (d) A head and a tail: [tex]\( \frac{1}{2} \)[/tex]
- (e) At most one tail: [tex]\( \frac{3}{4} \)[/tex]
