HG U8D3 Conditional Probability Dec3

Last updated 7 months ago

35 questions

Jada rolls one standard number cube, then she rolls another standard number cube.

What is the probability that she rolls a 5 on both number cubes? (answer as a fraction)

What is the probability that the second roll is a 5 under the condition that the first roll is a 6? (answer as a fraction)

What is the probability that the second roll is a 5 under the condition that the first roll is a 5? (answer as a fraction)

What is the probability that the second roll is not a 5? (answer as a fraction)

What is the probability that the first roll is a 5 and the second roll is not a 5? (answer as a fraction)

An Introduction to Conditional Probability
Imagine the last time you entered to win a raffle at a fair or carnival. You look at your ticket, 562104. As they begin to call off the winning ticket, you hear 562, but everyone has the same first 3 digits. Then 1 and 0 are called off. You know that excited feeling you get? Did you know there is a lot of math behind that instinct you feel that you might just win the prize? 

Now imagine those times when you are waiting to get your latest grade back on your English test. You’re not sure how you did, but as your teacher starts to talk about test results, her body language just isn’t positive. She keeps saying things like “Well, you guys tried hard.” 

Again, there is significant math happening behind that sinking feeling you now have. In this task, you will be investigating how probability can be used to formalize the way real-life conditions change the way we look at the world. 

Thinking about a winning raffle ticket, the fact that the probability in a given situation can change greatly affects how a situation is approached and interpreted. This sort of idea is prevalent across society, not just in games of chance. Knowledge of conditional probability can inform us about how one event or factor affects another.

Say-No-To-Smoking campaigns are vigilant in educating the public about the adverse health effects of smoking cigarettes. This motivation to educate the public has its beginnings in data analysis. Below is a table that represents a sampling of 500 people. Distinctions are made on whether or not a person is a smoker and whether or not they have ever developed lung cancer. Each number in the table represents the number of people that satisfy the conditions named in its row and column.

How does the table indicate that there is a connection between smoking and lung cancer?

Complete the table.

Using the 500 data points from the table, you can make reasonable estimates about the population at large by using probability. 500 data values are considered, statistically, to be large enough to draw conclusions about a much larger population. In order to investigate the table using probability, use the following outcomes:
S – The event that a person is a smoker
L – The event that a person develops lung cancer

Find each of the probabilities (write as a percentage):
P(S) = _______%

Find each of the probabilities (write as a percentage):
P(S') = _______% 

Find each of the probabilities (write as a percentage, round to the nearest tenth):
P(L) = _______% 

Find each of the probabilities (write as a percentage, round to the nearest tenth):
P(L') = _______ 

Find each of the probabilities (write as a percentage, round to the nearest tenth):
𝑃(𝐿 ∩ 𝑆)  = _______% 

Find each of the probabilities (write as a percentage, round to the nearest tenth):
𝑃(𝐿 ∩ 𝑆)' = _______% 

Find each of the probabilities (write as a percentage, round to the nearest tenth):
𝑃(𝐿 ∩ 𝑆') = _______% 

Find each of the probabilities (write as a percentage, round to the nearest tenth):
𝑃(𝐿' ∩ 𝑆) = _______ 

Find each of the probabilities (write as a percentage, round to the nearest tenth):
𝑃(𝑆 ∪ 𝐿) = _______ 

Find each of the probabilities (write as a percentage, round to the nearest tenth):
𝑃(𝑆' ∪ 𝐿') = _______ 

In order to use probability to reinforce the connection between smoking and lung cancer, you will use calculations of conditional probability.

By considering only those people who have been smokers, what is the probability of developing lung cancer?
_______ 

Compare the value above (in #18) to the one for P(L). What does this indicate?

You should be able to confirm that a non-smoker is less likely to develop lung cancer. By considering only non-smokers, what is the probability of developing lung cancer?

When calculating conditional probability, it is common to use the term “given.” In question 18, you have calculated the probability of a person developing lung cancer given that they are a smoker. The condition (or, “given”) is denoted with a single, vertical bar separating the probability needed from the condition. The probability of a person developing lung cancer given that they are a smoker is written 𝑃(𝐿|𝑆).

Rewrite the question from 18 using the word “given.”

Write the question from 18 using set notation.
_______ 

Find the probability that a person was a smoker given that they have developed lung cancer and represent it with proper notation. (write the probability as a fraction)

Find the probability that a given cancer-free person was not a smoker and represent it with proper notation. (write the probability as a fraction)

How does the probability in number 24 compare to 𝑃(𝐿|𝑆)? Are they the same or different and how so?

A Formula for Conditional Probability
The formulaic definition of conditional probability can be seen by looking at the different probabilities you calculated in part 2. The formal definition for the probability of event A given event B is the chance of both events occurring together with respect to the chance that B occurs. As a formula,

Movie executives collect lots of different data on the movies they show to determine who is going to see the different types of movies they produce. This will help them make decisions on a variety of factors from where to advertise a movie to what actors to cast. 

Below is a two-way frequency table that compares the preference of Harry Potter and the Deathly Hallows to Captain America: The First Avenger based upon the age of the moviegoer. 200 people were polled for the survey
Define each event in the table using the following variables: 
H – A person who prefers Harry Potter and the Deathly Hallows 
C – A person who prefers Captain America: The First Avenger 
Y – A person under the age of 30 
E – A person whose age is 30 or above

Complete the table.

Define each event in the table using the following variables:
H – A person who prefers Harry Potter and the Deathly Hallows
C – A person who prefers Captain America: The First Avenger
Y – A person under the age of 30
E – A person whose age is 30 or above

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
a.  P(E) = _______ 
b.  P(H) = _______ 
c.  P(C) = _______ 
d.  P(C|Y) = _______ 
e.  P(H|Y) = _______ 
f.  P(E|C) = _______ 
g. P(Y|C) = _______ 

Ice Cream
The retail and service industries are another aspect of modern society where probability’s relevance can be seen. By studying data on their own service and their clientele, businesses can make informed decisions about how best to move forward in changing economies. Below is a table of data collected over a weekend at a local ice cream shop, Frankie’s Frozen Favorites. The table compares a customer’s flavor choice to their cone choice.
C = Chocolate
B = Butter Pecan
F = Fudge Ripple
CC = Cotton Candy
S = Sugar Cone
W = Waffle Cone

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
P(W) = _______% 

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
P(B) = _______% 

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
P(S) = _______% 

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
P(F) = _______% 

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
P(S|C) = _______% 

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
P(S|B) = _______% 

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
P(S|CC) = _______% 

Find the following probabilities. (Use percent, rounded to the nearest tenth.)
P(B|W) = _______% 