Consider the following experiment: we toss a coin and throw a 6-faced die at the same time. The
coin is biased such that the probability of getting a head is 0.6. Similarly, we know that the die has a
special design such that
1. The probability of getting face 2 is twice that of getting face 1
2. The probability of getting face 3 is twice that of getting face 2
3. The probability of getting face 4 is twice that of getting face 3
4. The probability of getting face 5 is twice that of getting face 4
5. The probability of getting a face with odd numbers is twice that of getting a face with
even numbers
If we see the outcome of the die is an odd number, the reward obtained is just the number shown
on the face. Otherwise, the reward is twice the number shown on the face. Moreover, if the
outcome of the coin is a head, the reward obtained is 0.5. Otherwise, the reward is 1.




(Q2a) Please draw a bar chart to show the probability distribution of the total reward, where the
total reward is the sum of the rewards earned from the die and the coin? (1 mark)




(Q2b) What is the probability that the total reward is more than 4 and less than 12? (1 mark)





(Q2c) What is the variance of the total reward we can earn from this experiment? (1 mark)


Here is the bar chart showing the **probability distribution** of the total reward, which is the sum of the rewards earned from the die and the coin. The x-axis represents the possible total rewards, while the y-axis shows the corresponding probabilities.

### Step 3: Solve for Q2b and Q2c

#### (Q2b) **What is the probability that the total reward is more than 4 and less than 12?**
We need to calculate the probability for total rewards that fall in the range \( 4 < \text{reward} < 12 \).

#### (Q2c) **What is the variance of the total reward?**
We will compute the expected value (mean) and then use it to calculate the variance of the total rewards. Let's calculate these values next.### (Q2b) **Probability that the total reward is more than 4 and less than 12**:
The probability is approximately **0.578**.

### (Q2c) **Variance of the total reward**:
The variance of the total reward is approximately **6.34**.

Do you have any questions or need further details? Here are five related questions to extend the discussion:

1. How would the probability distribution change if the coin was fair (i.e., \( P(H) = P(T) = 0.5 \))?
2. How would the rewards and their probabilities change if the die had a different weighting for the faces?
3. What would be the effect on the variance if the coin's reward for heads was changed to 1 instead of 0.5?
4. Can you calculate the expected total reward if the die was fair?
5. How would the problem change if the reward for the die was based on a different function, such as a square of the face value?

**Tip:** When calculating variance, always remember that it's a measure of how spread out the values are from the mean.

Consider the following experiment: we toss a coin and throw a 6-faced die at the same time. The coin is biased such that the probability of getting a head is 0.6. Similarly, we know that the die has a special design such that the probabilities of each face are not uniform. The reward depends on both the coin and die outcomes. (Q2a) Draw a bar chart to show the probability distribution of the total reward. (Q2b) What is the probability that the total reward is more than 4 and less than 12? (Q2c) What is the variance of the total reward?

Explore the probability distribution, expected value, and variance of a total reward from an experiment involving a biased coin and a specially weighted die. This problem covers key concepts in probability theory and statistical analysis. Suitable for high school students in Grades 9-12.

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