# Probabilities of Picking Correct Cases

## Example

There are five matches on each weekend of a basketball season. Ben takes part in a competition where he earns one point if he picks more than half of the winning teams for a weekend and zero points otherwise. The probability that Ben correctly picks the team that wins any given match is \( 0.7 \).

## Part 1

Find the probability that Ben earns one point for a given weekend.

\( \displaystyle \begin{align} \Pr(\text{correct pick}) &= 0.7 \\ \Pr(\text{incorrect pick}) &= 0.3 \\ \Pr(\text{One point}) &= \Pr(\text{3, 4 or 5 winning teams are picked}) \\ &= \Pr(\text{3 correct picks}) + \Pr(\text{4 correct picks}) + \Pr(\text{5 correct picks}) \\ &= {5 \choose 3} 0.3^2 \times 0.7^3 + {5 \choose 4} 0.3^1 \times 0.7^4 + {5 \choose 5} 0.3^0 \times 0.7^5 \\ &= 0.83692 \end{align} \)

## Part 2

Hence, find the probability that Ben earns one point for a given fifteen-week season. Give your answer correct to one significant figure.

\( \displaystyle \begin{align} \Pr(\text{One point for 15 weeks}) &= 0.83692^{15} \\ &= 0.069224 \cdots \\ &= 0.07 \ \text{(one significant figure)} \end{align} \)

## Part 3

Find the probability that ben earns at most 13 points during the fifteen-week season. Give your answer correct to two significant figures.

\( \displaystyle \begin{align} \Pr(\text{at most 13 points}) &= \Pr(\text{0 or 1 or 2 or } \cdots \text{ or 13 points}) \\ &= 1-\Pr(\text{14 points})-\Pr(\text{15 points}) \\ &= 1-{15 \choose 14} 0.16308^1 \times 0.83692^{14}-{15 \choose 15} 0.16308^0 \times 0.83692^{15} \\ &= 0.72844 \cdots \\ &= 0.73 \ \text{(two significant figures)} \end{align} \)

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

## Responses