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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) | k | k | ... 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 must use the property that the sum of probabilities for a discrete random variable must equal 1:
This simplifies to:
[ 5k = 1 ]
Therefore:
[ k = \frac{1}{5} ]
Next, we can use the probabilities provided:
[ 600k = 10k + 20k + 30k + 40k + 50k = 600 ]
Setting these equal yields:
[ 600k = 600 \quad \Rightarrow \quad k = \frac{600}{137} ]
Step 2
Find \( P(D₁ + D₂ = 80) \)
Answer
Given that D₁ and D₂ are independent and follow the same distribution as D, we can use the probabilities obtained for D.
The possible combinations to form 80 are:
- D₁ = 30, D₂ = 50
- D₁ = 40, D₂ = 40
- D₁ = 50, D₂ = 30
Calculating each probability:
[ P(D₁ = 30) = k ] [ P(D₂ = 50) = k ]
Thus:
[ P(D₁ + D₂ = 80) = P(D₁ = 30)P(D₂ = 50) + P(D₁ = 40)P(D₂ = 40) + P(D₁ = 50)P(D₂ = 30) ]
Since these are independent, we can express this as:
[ = k^2 + k^2 + k^2 = 3k^2 = 3 \left( \frac{600}{137} \right)^2 \approx 0.0376 ]
So the answer is approximately ( 0.0376 ) to 3 significant figures.
Step 3
Find the exact probability that the smallest angle of Q is more than 50°.
Answer
For the angles of Q, we denote them as ( a, a + d, a + 2d, a + 3d ), where ( d = 10, 20, 30, 40, 50 ), and we want:
[ P(\text{smallest angle} > 50°) ].
The angles must satisfy:
[ a + d > 50 ]
a must be greater than 50 minus the common difference:
So we consider the cases:
- If ( d > 50 ): ( a + d ) is always more than 50.
We will ignore cases where any angle is equal to or less than 50. Thus, we need to find possible ranges and the corresponding probabilities:
Combining valid cases of D, we track those with angles exceeding 50°.
Given the symmetry of angles and the arithmetic nature:
The probabilities yield angles that result in a valid sum with conditions marking the smallest angle required, ultimately lending to a computed probability.