\(A\) and \(B\) are independent events.... | Mathematics 0580 | Paper 4 Mock 01 | Calculator (Extended)

Question 1 Report

\(A\) and \(B\) are independent events.

\(\text{P}(A) = x\), \(\text{P}(B) = x + 0.2\) and \(\text{P}(A \text{ and } B) = 0.15\).

Find the value of \(x\).

Answer

For independent events the probability that both occur is the product of the separate probabilities:

\[\text{P}(A\text{ and }B)=\text{P}(A)\times\text{P}(B)\]

Substituting the given expressions turns a probability statement into a quadratic equation:

\[x(x+0.2)=0.15\]

[M1]

Expand and collect everything on one side:

\[x^{2}+0.2x-0.15=0\]

[M1]

Factorising, two numbers with product \(-0.15\) and sum \(0.2\) are \(-0.3\) and \(0.5\):

\[(x-0.3)(x+0.5)=0\]

[M1] oe (formula)

The two roots are \(x=0.3\) and \(x=-0.5\), but a probability cannot be negative, so

\[x=0.3\]

[A1] (\(x=-0.5\) rejected)

Rejecting the negative root with a brief reason is part of the answer, not an optional extra. Check the solution in context: \(\text{P}(A)=0.3\) and \(\text{P}(B)=0.5\), both valid probabilities, and \(0.3\times 0.5=0.15\) as required. The quadratic formula gives the same roots if the factorisation is not spotted, and multiplying the equation by 100 first, to give \(100x^{2}+20x-15=0\), avoids decimals if you prefer whole numbers.

Read lesson note on Probability Of Combined Events (CIE)

Answer Details

For independent events the probability that both occur is the product of the separate probabilities:

\[\text{P}(A\text{ and }B)=\text{P}(A)\times\text{P}(B)\]

Substituting the given expressions turns a probability statement into a quadratic equation:

\[x(x+0.2)=0.15\]

[M1]

Expand and collect everything on one side:

\[x^{2}+0.2x-0.15=0\]

[M1]

Factorising, two numbers with product \(-0.15\) and sum \(0.2\) are \(-0.3\) and \(0.5\):

\[(x-0.3)(x+0.5)=0\]

[M1] oe (formula)

The two roots are \(x=0.3\) and \(x=-0.5\), but a probability cannot be negative, so

\[x=0.3\]

[A1] (\(x=-0.5\) rejected)

Read lesson note on Probability Of Combined Events (CIE)

Probability Of Combined Events

View Answer

\(A\) and \(B\) are independent events. \(\text{P}(A) = x\), \(\text{P}(B) = x + 0.2\) and \(\text{P}(A \text{ and } B) = 0.15\). Find the value of \(x\).

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