To solve this problem, let’s analyze each question using the given probability distribution graph. The graph consists of two line segments and a curved path. The probability distribution function (PDF) shows the probability density across different intervals on the x-axis.

(a) Probability that the number picked is between 4.7 and 5.3

Since the probability density function is given, we can find the probability over an interval by calculating the area under the curve between the limits 4.7 and 5.3.

From the graph:

• Between 4 and 6, the PDF is constant at 0.25.
• The interval [4.7, 5.3] is within [4, 6], so the probability density in this range is also 0.25.

To find the probability, we calculate: $\text{Probability} = \text{density} \times \text{width of interval} = 0.25 \times (5.3 - 4.7) = 0.25 \times 0.6 = 0.15.$

Answer for (a): $0.15$.

(b) Probability that the number picked is between 6 and 9

To find this probability, we need the area under the curve between x = 6 and x = 9.

From the graph:

• The PDF has a straight-line segment that decreases from 0.25 at x = 6 to 0 at x = 9.
• This region forms a triangle with a base of $9 - 6 = 3$ and a height of 0.25.

The area of a triangle is calculated as: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 0.25 = 0.375.$

Answer for (b): $0.375$.

(c) Probability that the number picked is 5

In continuous probability distributions, the probability of picking any exact value (like exactly 5) is zero. This is because the area under the curve at a single point is zero.

Answer for (c): $0$.

Would you like further details or have any questions?

Here are some related questions:

1. What is the probability that the number picked is between 2 and 4?
2. How would you calculate the probability for an interval outside the given distribution, like between 0 and 2?
3. What is the total probability over the interval from 2 to 9?
4. How does the probability change if the interval is halved in size?
5. Can you calculate the mean of this distribution based on the PDF?

Tip: For continuous distributions, remember that the probability of selecting an exact point is always zero.