A lecture hall has 200 seats with folding arm tablets, 30 of...

A lecture hall has 200 seats with folding arm tablets, 30 of which are designed for left handers. The typical size of classes that meet there is $188,$ and we can assume that about 13$\%$ of students are left-handed. What's the probability that a right-handed student in one of these classes is forced to use a lefty arm tablet?

A lecture hall has 200 seats with folding arm tablets, 30 of which are designed for left handers. The
typical size of classes that meet there is $188,$ and we can assume that about 13$\%$ of students are left-handed. What's the probability that a right-handed student in one of these classes is forced to use a lefty arm tablet?

Key Concepts

Binomial Distribution

This concept models the number of occurrences of a particular outcome in a fixed number of independent trials with a constant probability for each trial. In contexts like this, where a given proportion of individuals (e.g., left?handers) is expected among a fixed group (such as a class), the binomial distribution provides the mathematical framework to assess the likelihood of various counts of that outcome.

Conditional Probability

Conditional probability focuses on determining the probability of an event given that another event has occurred. In seating assignment problems, for example, it is used to calculate the chance that a right-handed student is forced to use a left-handed seat based on the realized number of left-handed students, which directly affects the allocation of seating resources.

Occupancy Problems in Combinatorics

Occupancy or allocation problems involve distributing a set of objects into containers under specific constraints, such as limited capacity or preference rules. This type of problem often appears when assigning students to seats where some seats have special features, and not everyone can be accommodated in the preferred seat type. It provides a structure for analyzing conflicts between supply (seats) and demand (students with preferences).

Transcript

00:01 In this exercise, we are given a scenario where there's a lecture hall with 200 seats that have folding arm tablets, 30 of which are designed for left -handers.

00:09 And we're also told that the typical class size in that lecture hall is 188 people.

00:15 And we can assume that 13 % of students are left -handed.

00:19 And we are asked to calculate the probability that a right -handed student in one of the classes must use the lefty arm tablet.

00:27 So there's a few numbers here.

00:29 So we have an auditorium with 200 seats.

00:32 It is important to note that our sample size is 188 because that's the number of people that typically go to this class.

00:41 If we're looking for a situation where a right -hander must use a left -handed seat or arm tablet, that means there has to be more right -handed people than there are right -handed arm tablets.

00:55 We're told there are 200 seats and 30 are designed for left -handers, which means that we have 200 minus 30 or 70 designed for right -handers.

01:08 It's 170.

01:13 So it seems easier to approach this problem by defining a success as a student being right -handed, in which case the probability success is 100 minus 13%, which is 0 .87.

01:28 And we can consider each student to be a bernoulli trial, where there's success or failure.

01:33 Success is being right -handed.

01:34 The probability of success is constant at 0 .87, and each student's outcome is independent from the other students.

01:47 So if we can assume that these are bernoulli trials, and we define the random variable x as the number of right -handers in 188 students, then x is a binomial random variable based on 188 trials and probability of success of 0 .87.

02:22 Computationally it will be easier to express this or to approximate this with the normal distribution.

02:28 So first we want to check to see if n times p and n times q are both at least as big as 10...

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