Answer :
Sure, let's solve the given problems step-by-step, using a detailed explanation for each part.
### Part 1: Two Dice are Thrown
#### (a) Possibility Diagram
The dice can take values from 1 to 6, so the possible outcomes of throwing two dice are:
```
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
```
There are a total of 36 possible outcomes (6 outcomes for the first die × 6 outcomes for the second die).
#### (b) Probabilities
(i) Probability that the total score is 5
- The total score of 5 can be obtained with the pairs: (1,4), (2,3), (3,2), (4,1)
- Number of favorable outcomes: 4
- Hence, the probability is [tex]\(\frac{4}{36} = \frac{1}{9} \approx 0.1111\)[/tex] (exactly 0.1111111111111111)
(ii) Probability that the total score is 10
- The total score of 10 can be obtained with the pairs: (4,6), (5,5), (6,4)
- Number of favorable outcomes: 3
- Hence, the probability is [tex]\(\frac{3}{36} = \frac{1}{12} \approx 0.0833\)[/tex] (exactly 0.08333333333333333)
(iii) Probability that the total score is greater than 6
- Pairs with a total score greater than 6 include: (1,6), (2,5), (2,6), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
- Number of favorable outcomes: 21
- Hence, the probability is [tex]\(\frac{21}{36} = \frac{7}{12} \approx 0.5833\)[/tex] (exactly 0.5833333333333334)
(iv) Probability that the total score is a square number
- Possible square numbers are 1, 4, 9, 16, 25, 36. Only 4 and 9 are possible as sums of two dice.
- Pairs with a total score of 4: (1,3), (2,2), (3,1)
- Pairs with a total score of 9: (3,6), (4,5), (5,4), (6,3)
- Number of favorable outcomes: 7
- Hence, the probability is [tex]\(\frac{7}{36} \approx 0.1944\)[/tex] (exactly 0.19444444444444445)
### Part 2: A Coin is Tossed and a Die is Thrown
#### (a) Possibility Diagram
The coin can show H (Heads) or T (Tails), and the die can show values from 1 to 6, so the possible outcomes are:
```
(H,1) (H,2) (H,3) (H,4) (H,5) (H,6)
(T,1) (T,2) (T,3) (T,4) (T,5) (T,6)
```
There are a total of 12 possible outcomes (2 outcomes for the coin × 6 outcomes for the die).
#### (b) Probabilities
(i) Probability of getting a two and a head
- The outcome is (H,2)
- Number of favorable outcomes: 1
- Hence, the probability is [tex]\(\frac{1}{12} \approx 0.0833\)[/tex] (exactly 0.08333333333333333)
(ii) Probability of getting a tail and an odd number
- Possible outcomes are: (T,1), (T,3), (T,5)
- Number of favorable outcomes: 3
- Hence, the probability is [tex]\(\frac{3}{12} = \frac{1}{4} = 0.25\)[/tex]
(iii) Probability of getting a head and a prime number
- Prime numbers less than or equal to 6 are 2, 3, 5.
- Possible outcomes are: (H,2), (H,3), (H,5)
- Number of favorable outcomes: 3
- Hence, the probability is [tex]\(\frac{3}{12} = \frac{1}{4} = 0.25\)[/tex]
### Part 3: Two Coins are Tossed
#### Possibility Diagram
The outcomes for tossing two coins are:
```
(H,H) (H,T) (T,H) (T,T)
```
There are a total of 4 possible outcomes.
#### Probabilities
(a) Probability that both coins show a tail
- The outcome is (T,T)
- Number of favorable outcomes: 1
- Hence, the probability is [tex]\(\frac{1}{4} = 0.25\)[/tex]
(b) Probability that at least one coin shows a tail
- Possible outcomes are: (H,T), (T,H), (T,T)
- Number of favorable outcomes: 3
- Hence, the probability is [tex]\(\frac{3}{4} = 0.75\)[/tex]
(c) Probability that neither coin shows a tail
- The outcome is (H,H)
- Number of favorable outcomes: 1
- Hence, the probability is [tex]\(\frac{1}{4} = 0.25\)[/tex]
(d) Probability that either coin shows a tail
- This is equivalent to the probability of at least one tail.
- Hence, the probability is [tex]\(\frac{3}{4} = 0.75\)[/tex]
I hope this explanation helps you understand how to arrive at the probabilities for the given scenarios! If you have any more questions, feel free to ask.
### Part 1: Two Dice are Thrown
#### (a) Possibility Diagram
The dice can take values from 1 to 6, so the possible outcomes of throwing two dice are:
```
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
```
There are a total of 36 possible outcomes (6 outcomes for the first die × 6 outcomes for the second die).
#### (b) Probabilities
(i) Probability that the total score is 5
- The total score of 5 can be obtained with the pairs: (1,4), (2,3), (3,2), (4,1)
- Number of favorable outcomes: 4
- Hence, the probability is [tex]\(\frac{4}{36} = \frac{1}{9} \approx 0.1111\)[/tex] (exactly 0.1111111111111111)
(ii) Probability that the total score is 10
- The total score of 10 can be obtained with the pairs: (4,6), (5,5), (6,4)
- Number of favorable outcomes: 3
- Hence, the probability is [tex]\(\frac{3}{36} = \frac{1}{12} \approx 0.0833\)[/tex] (exactly 0.08333333333333333)
(iii) Probability that the total score is greater than 6
- Pairs with a total score greater than 6 include: (1,6), (2,5), (2,6), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
- Number of favorable outcomes: 21
- Hence, the probability is [tex]\(\frac{21}{36} = \frac{7}{12} \approx 0.5833\)[/tex] (exactly 0.5833333333333334)
(iv) Probability that the total score is a square number
- Possible square numbers are 1, 4, 9, 16, 25, 36. Only 4 and 9 are possible as sums of two dice.
- Pairs with a total score of 4: (1,3), (2,2), (3,1)
- Pairs with a total score of 9: (3,6), (4,5), (5,4), (6,3)
- Number of favorable outcomes: 7
- Hence, the probability is [tex]\(\frac{7}{36} \approx 0.1944\)[/tex] (exactly 0.19444444444444445)
### Part 2: A Coin is Tossed and a Die is Thrown
#### (a) Possibility Diagram
The coin can show H (Heads) or T (Tails), and the die can show values from 1 to 6, so the possible outcomes are:
```
(H,1) (H,2) (H,3) (H,4) (H,5) (H,6)
(T,1) (T,2) (T,3) (T,4) (T,5) (T,6)
```
There are a total of 12 possible outcomes (2 outcomes for the coin × 6 outcomes for the die).
#### (b) Probabilities
(i) Probability of getting a two and a head
- The outcome is (H,2)
- Number of favorable outcomes: 1
- Hence, the probability is [tex]\(\frac{1}{12} \approx 0.0833\)[/tex] (exactly 0.08333333333333333)
(ii) Probability of getting a tail and an odd number
- Possible outcomes are: (T,1), (T,3), (T,5)
- Number of favorable outcomes: 3
- Hence, the probability is [tex]\(\frac{3}{12} = \frac{1}{4} = 0.25\)[/tex]
(iii) Probability of getting a head and a prime number
- Prime numbers less than or equal to 6 are 2, 3, 5.
- Possible outcomes are: (H,2), (H,3), (H,5)
- Number of favorable outcomes: 3
- Hence, the probability is [tex]\(\frac{3}{12} = \frac{1}{4} = 0.25\)[/tex]
### Part 3: Two Coins are Tossed
#### Possibility Diagram
The outcomes for tossing two coins are:
```
(H,H) (H,T) (T,H) (T,T)
```
There are a total of 4 possible outcomes.
#### Probabilities
(a) Probability that both coins show a tail
- The outcome is (T,T)
- Number of favorable outcomes: 1
- Hence, the probability is [tex]\(\frac{1}{4} = 0.25\)[/tex]
(b) Probability that at least one coin shows a tail
- Possible outcomes are: (H,T), (T,H), (T,T)
- Number of favorable outcomes: 3
- Hence, the probability is [tex]\(\frac{3}{4} = 0.75\)[/tex]
(c) Probability that neither coin shows a tail
- The outcome is (H,H)
- Number of favorable outcomes: 1
- Hence, the probability is [tex]\(\frac{1}{4} = 0.25\)[/tex]
(d) Probability that either coin shows a tail
- This is equivalent to the probability of at least one tail.
- Hence, the probability is [tex]\(\frac{3}{4} = 0.75\)[/tex]
I hope this explanation helps you understand how to arrive at the probabilities for the given scenarios! If you have any more questions, feel free to ask.
