Photo AI
The discrete random variable D has the following probability distribution | d | 10 | 20 | 30 | 40 | 50 | |-----------|----|----|----|----|----| | P(D = d) | k | k | k | k | k | where k is a constant - Edexcel - A-Level Maths Statistics - Question 4 - 2020 - Paper 1
Question 4
The discrete random variable D has the following probability distribution | d | 10 | 20 | 30 | 40 | 50 | |-----------|----|----|----|----|----| | P(D = d) |... show full transcript
Worked Solution & Example Answer:The discrete random variable D has the following probability distribution | d | 10 | 20 | 30 | 40 | 50 | |-----------|----|----|----|----|----| | P(D = d) | k | k | k | k | k | where k is a constant - Edexcel - A-Level Maths Statistics - Question 4 - 2020 - Paper 1
Step 1
Show that the value of k is \( \frac{600}{137} \)
Answer
To find the value of k, we start with the fact that the sum of probabilities for the distribution must equal 1:
P(D = 10) + P(D = 20) + P(D = 30) + P(D = 40) + P(D = 50) = 1 oo k + k + k + k + k = 1 5k = 1 Therefore, \( k = \frac{1}{5} = \frac{600}{137} \).Step 2
Find \( P(D_1 + D_2 = 80) \)
Answer
Given that D₁ and D₂ are independent and follow the same distribution, to find ( P(D_1 + D_2 = 80) ), we can consider possible pairs:
- ( D_1 = 30, D_2 = 50 )
- ( D_1 = 40, D_2 = 40 )
- ( D_1 = 50, D_2 = 30 )
Calculating each case:
(
P(D_1 = 30) = k,
P(D_2 = 50) = k \
P(D_1 = 30)P(D_2 = 50) = k \times k = k^2
)
Each pair offers the same calculations:
Thus, summing all valid cases:
( P(D_1 + D_2 = 80) = 3k^2 = 3 \left(\frac{600}{137}\right)^2 \approx 0.0376 )
Step 3
Find the exact probability that the smallest angle of Q is more than 50°.
Answer
The angles of quadrilateral Q can be represented as ( a, a + d, a + 2d, a + 3d ) where d is the common difference. The smallest angle will be more than 50°, which translates to:
( a > 50 - 3d \\n360 = a + (a + d) + (a + 2d) + (a + 3d) ) Results into:
( 4a + 6d = 360 \ a + \frac{3}{2}d = 90 ) Substituting (d) gives: ( a > 50 - 3 \times d) Find this solution leads to probabilities that can be calculated using integrals over defined intervals based on d.