// Numbas version: exam_results_page_options {"name": "Calculating expected values given a table of probabilities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Calculating expected values given a table of probabilities", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

This question assesses the students ability to find the expected number of times an event occurs given the probability of the event occurring for a single trial and the total number of trials.

"}, "statement": "

The were 542 reported injuries which caused footballers to miss either games or days of training during the 2017 Premier League season.

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The table below shows the probability that an injured player suffered a particular injury, denoted $P(\\text{Injury})$.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 Injury $P(\\text{Injury})$ Structure Hamstring $\\var{Hamstring}$ Muscle Knee $\\var{Knee}$ Joint Ankle $\\var{Ankle}$ Joint Illness $\\var{Illness}$ Immune System
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A random sample of $\\var{no_people}$ of these injured or ill athletes attended your clinic.

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\$\\text{Expected number of times an event occurs} = \\text{Probability of event} \\times \\text{Number of trials}.\$

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a)

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The expected number of people attending the clinic who have had a knee injury is

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\$\\text{probabilty of having a knee injury} \\times \\text{number of people attending clinic} = \\var{Knee} \\times \\var{no_people} = \\var{{Knee}*{no_people}}. \$

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b)

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Let us denote the probability that a person has damaged a joint $P(\\text{Joint})$. We can see that there are two injuries which belong to the Joint structure: Knee and Ankle. Therefore, it is true that

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\\\begin{align} P(\\text{Joint}) &= P(\\text{Knee})+P(\\text{Ankle})\\\\ &= \\var{Knee}+\\var{Ankle}\\\\ &= \\var{Knee+Ankle}. \\end{align} \

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Then the expected number of people who have damaged a joint is

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\$\\var{Knee + Ankle} \\times \\var{no_people} = \\var{dpformat(({Knee + Ankle})*{no_people},2)}, \$

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which rounds to $\\var{dpformat(({Knee + Ankle})*{no_people},0)}$ to the nearest integer.

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How many athletes would you expect to have had a knee injury?

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How many athletes would you have expected to damaged a joint?

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Number of people who see a movie.

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