A fair blue die has faces numbered 1, 1, 3, 3 and 5 - Edexcel - A-Level Maths Statistics - Question 6 - 2013 - Paper 1
Question 6
A fair blue die has faces numbered 1, 1, 3, 3 and 5. The random variable B represents the score when the blue die is rolled.
(a) Write down the probability distribu... show full transcript
Worked Solution & Example Answer:A fair blue die has faces numbered 1, 1, 3, 3 and 5 - Edexcel - A-Level Maths Statistics - Question 6 - 2013 - Paper 1
Step 1
Write down the probability distribution for B.
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Answer
The probability distribution for the random variable B is as follows:
b
1
3
5
P(B = b)
2/5
2/5
1/5
Step 2
State the name of this probability distribution.
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Answer
The name of this probability distribution is 'Discrete Uniform'.
Step 3
Write down the value of E(B).
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Answer
The expected value of B, E(B), can be calculated using the formula:
E(B) = rac{1}{n} imes ext{sum of all values}
In this case:
E(B) = rac{1}{5}(1 + 1 + 3 + 3 + 5) = rac{13}{5} = 2.6
Step 4
Find E(R).
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Answer
To find the expected value E(R), we apply the formula:
E(R)=extsumof(rimesextP(R=r))
Calculating:
E(R) = 2 imes rac{2}{3} + 4 imes rac{1}{6} + 6 imes rac{1}{6}E(R) = rac{4}{3} + rac{4}{6} + rac{6}{6} = rac{4}{3} + rac{2}{3} + 1 = rac{11}{3}
Step 5
Find Var(R).
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Answer
To find the variance Var(R), we use the formula:
Var(R)=E(R2)−(E(R))2
First, we compute E(R^2):
E(R^2) = 2^2 imes rac{2}{3} + 4^2 imes rac{1}{6} + 6^2 imes rac{1}{6}
Calculating this gives:
E(R^2) = rac{8}{3} + rac{16}{6} + rac{36}{6} = rac{8}{3} + rac{26}{6}
Now, plugging into the variance formula:
First convert to a common denominator:
Var(R) = E(R^2) - rac{121}{9}
Step 6
Find the probability that Avisha wins the game, stating clearly which die she should use in each case.
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Answer
When the coin shows 2:
Avisha would roll the blue die (B) which has the probability of:
P(B > 2) = P(B = 3) + P(B = 3) + P(B = 5) = rac{2}{5} + rac{1}{5} = rac{3}{5}
When the coin shows 5:
Avisha would roll the red die (R) since all values are less:
P(R>5)=0
Thus, the overall winning probability for Avisha is:
P(Avisha ext{ wins}) = rac{1}{2} imes rac{3}{5} + rac{1}{2} imes 0 = rac{3}{10}
