How many of these people would you expect to have bought tickets to see *Forgotten Game*?

How many of these people would you expect to have bought tickets to see a Sci-Fi film?

", "showCorrectAnswer": true, "scripts": {}, "correctAnswerStyle": "plain", "marks": 1}], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "variables": {"Avatar": {"group": "Ungrouped variables", "description": "Probability someone sees Avatar

", "name": "Avatar", "templateType": "anything", "definition": "random(0.2..0.31 #0.05)"}, "NYSM": {"group": "Ungrouped variables", "description": "Probability someone goes to see Now you see me

", "name": "NYSM", "templateType": "anything", "definition": "(1-(Avatar+SW))*3/5"}, "no_people": {"group": "Ungrouped variables", "description": "Number of people who see a movie.

", "name": "no_people", "templateType": "anything", "definition": "random(100..180 #20)"}, "SW": {"group": "Ungrouped variables", "description": "Probability someone goes to see Star Wars

", "name": "SW", "templateType": "anything", "definition": "random(0.4..0.51 #0.05)"}, "TIJ": {"group": "Ungrouped variables", "description": "Probability someone goes to see the Italian Job

", "name": "TIJ", "templateType": "anything", "definition": "1-(Avatar+SW+NYSM)"}}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "rulesets": {}, "variable_groups": [], "name": "Calculating Expected Values given a table of probabilities", "preamble": {"js": "", "css": ""}, "advice": "If we are given the probability of an event occurring in a single trial then we can calculate the expected number of times that this event would occur in a larger number of trials.

\nTo do this, we multiply the probability of the event occurring in a single trial by the total number of trials:

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

\nWe are given the probabilities that someone buys a ticket to see each film in the table below.

\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\nFilm | $P(\\text{Film})$ | Genre |

Forgotten Game | $\\var{Avatar}$ | Sci-Fi |

The Diamond Valley | $\\var{SW}$ | Sci-Fi |

School of Return | $\\var{NYSM}$ | Thriller |

The Silk's Nobody | $\\var{TIJ}$ | Crime |

We are also told that $\\var{no_people}$ people each buy a ticket at the cinema to see a film of their own choosing during this day.

\nTo calculate the expected number of people who bought tickets to see one of these films we multiply the probability that a person buys a ticket for that film by how many people bought tickets for a film at the cinema.

\nSo the expected number of people who bought tickets to see *Forgotten Game* is

\\[

\\var{Avatar} \\times \\var{no_people} = \\var{{Avatar}*{no_people}}.

\\]

We are now asked to calculate the expected number of people who bought tickets to see a Sci-Fi film.

\nFrom the table above we can see that there are two films which belong to the Sci-Fi genre: *Forgotten Game* and *The Diamond Valley*.

Firstly, we need to calculate the probability that a person buys a ticket to see a Sci-Fi film, which we will denote $P(\\text{Sci-Fi})$.

\nSince the probability that a person buys a ticket to see each film is different, it would be **incorrect** to say that the probability that a person buys a ticket to see a Sci-Fi film is

\\[\\displaystyle\\frac{2}{4} = \\displaystyle\\frac{1}{2}.\\]

\nInstead we must recognise that the probability that a person buys a ticket to see a Sci-Fi film is the probability that a person buys a ticket to see either *Forgotten ***or** *The Diamond Valley*.

Therefore to calculate this probability, we add the probabilities of a person buying a ticket to see each of these films:

\n\\[

\\begin{align}

P(\\text{Sci-Fi}) &= P(\\text{Forgotten Game})+P(\\text{The Diamond Valley})\\\\

&= \\var{Avatar}+\\var{SW}\\\\

&= \\var{Avatar+SW}.

\\end{align}

\\]

Then the expected number of people who bought tickets to see a Sci-Fi film is

\n\\[

\\var{Avatar+SW} \\times \\var{no_people} = \\var{({Avatar+SW})*{no_people}}.

\\]

There are four films being shown in a cinema on a particular day.

\nThe probability that a person buys a ticket to see each film, denoted $P(\\text{Film})$, is given in the table below.

\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\nFilm | $P(\\text{Film})$ | Genre |

Forgotten Game | $\\var{Avatar}$ | Sci-Fi |

The Diamond Valley | $\\var{SW}$ | Sci-Fi |

School of Return | $\\var{NYSM}$ | Thriller |

The Silk's Nobody | $\\var{TIJ}$ | Crime |

$\\var{no_people}$ people each buy a ticket at the cinema to see a film of their own choosing during the day.

", "functions": {}, "ungrouped_variables": ["Avatar", "SW", "NYSM", "TIJ", "no_people"], "tags": ["Dice", "dice", "Expected values", "Expected Values", "Experimental Probability", "experimental probability", "Experimental probability", "Probability", "probability", "relative frequency", "Relative Frequency", "taxonomy", "Theoretical Probability", "theoretical probability"], "metadata": {"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.

", "licence": "Creative Commons Attribution 4.0 International"}}], "pickingStrategy": "all-ordered"}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}]}