Unit 8 Family Materials
Conditional Probability
Conditional Probability
In this unit, your student will build on their understanding of probability, including conditional probability. The probability of an event is a number that measures how likely the event is to happen. It can be 0, 1, or any number in between. It is 0 if the event will never happen and 1 if the event must happen. If an event occurs half of the time in the long run, then its probability is 0.5. Conditional probability is the probability that one event occurs under the condition that another event occurs.
Here is an example. The table summarizes the type (medium, large, or extralarge) and condition (no cracked eggs, or one or more cracked eggs) of 50 cartons of eggs at a grocery store.
medium  large  extralarge  total  

one or more eggs cracked  1  3  1  10 
no cracked eggs  4  22  19  40 
total  5  25  20  50 
One carton is selected at random.
What is the probability that the carton has no cracked eggs? This probability is 0.8. This is because 40 cartons have no cracked eggs out of a total of 50 cartons and \(\frac{40}{50} = 0.8\). Students also see this type of of question written as \(P(\text{no cracked eggs})\) which means “the probability that a randomly selected carton has no cracked eggs.” In this case, \(P(\text{no cracked eggs}) = 0.8\).
What is the probability that the carton has no cracked eggs under the condition that it is a carton of extralarge eggs? This conditional probability is 0.95. This is because 19 cartons of extralarge eggs had no cracked eggs out of a total of 20 cartons of extralarge eggs and \(\frac{19}{20} = 0.95\). This type of probability is called conditional probability because it is a probability based on the condition of selecting a carton of extralarge eggs. Students see this type of question written as \(P(\text{no cracked eggs})\) which means that the “probability that a randomly selected carton has no cracked eggs under the condition that it is a carton of extralarge eggs.” In this case \(P(\text{no cracked eggs  extralarge}) = 0.95\).
Here is a task to try with your student:
The table summarizes the position of loaves of bread at the grocery store (bread in the front row or bread not in the front row) and the sellby date (within five days or more than 5 days away).
A loaf of bread is selected at random.
sellby date within 5 days  sellby date more than 5 days away  

bread in the front row  36  14 
bread not in the front row  24  76 

What is the probability that the bread has a sellby date within 5 days?

What is the probability that bread has a sellby date within 5 days under the condition that the loaf of bread is in the front row?

What is \(P(\text{sellby date more than 5 days away  bread not in the front row})\)?

You are in a rush and want to grab a loaf of bread at this store without looking at the sellby date. Does grabbing the loaf of bread from the front row give you the best chance of getting a loaf of bread with a sellby date more than 5 days away? Use probability to explain your reasoning.
Solution:

0.4 or \(\frac{60}{150}\)

0.72 or \(\frac{36}{50}\)

0.76 or \(\frac{76}{100}\)

No it does not give you the best chance of getting a loaf of bread with a sellby date more than 5 days away. The probability of getting a loaf or bread with a sellby date more than 5 days away under the condition that is in the front row is 0.28 compared to a probability of 0.72 for a loaf of bread that is not in the front row.