Probability

# Conditional Probability: Level 2 Challenges

There are three cards on a table. One of them is black on both sides, one of them is black on one side and red on the other, and one of them is red on both sides.

One of these three cards is selected at random and placed on a second table with a side randomly chosen to face up. The side facing upward is red.

What is the probability that the side facing the table is also red?

You flip a coin and roll a die. Let $H$ be the event you flip a heads and let $F$ be the event that you roll a 4. What is $P\left(H\ | \ F\right)?$

Note: $P\left(H\ | \ F\right)$ denotes the probability of $H$ occurring given that $F$ occurs.

You randomly choose a treasure chest to open, and then randomly choose a coin from that treasure chest. If the coin you choose is gold, then what is the probability that you chose chest A?

1% of people have a rare cancer, and there is a test for this cancer which is "90% accurate"; that is:

• If you have the cancer there is a 90% chance the test will be positive.

• If you don't have the cancer there is a 90% chance the test will be negative.

If you take the test and test positive, what is the approximate probability that you have the cancer?

In the Monty Hall Quiz show, Monty offers the successful candidate another challenge to determine what prize he/she gets. Monty shows the contestant 3 doors, behind one door there is a new car and behind the other two doors there are goats. The contestant chooses one door and then Monty opens one of the other doors which he knows contains a goat.

He then offers the contestant the chance of changing his choice. Should the contestant switch to the other unopened door?

A. No, he shouldn't because he should trust his first instinct.
B. No, he shouldn't because Monty is trying to trick him.
C. Yes, he should because the odds have changed.
D. Maybe, but his odds haven't changed.

10% of all items produced are defective. The worker who chooses items for inspection has a sense for which are defective, so 60% of all defective items are inspected, while only 20% of all good items are inspected. If an item is inspected, what is the probability it is defective?