Dependent probability introduction | Probability and Statistics | Khan Academy

TL;DR

In this video, the probability of winning a game involving marbles is calculated, and it is determined that it is not economically beneficial to play.

Transcript

Let's imagine ourselves in some type of a strange casino with very strange games. And you walk up to a table, and on that table there is an empty bag. And the guy who runs the table says, look, I've got some marbles here, 3 green marbles, 2 orange marbles. And I'm going to stick them in the bag. And he literally sticks them into the empty bag to sh... Read More

Key Insights

• ⏮️ The probability of an event depends on whether the event is independent or dependent on previous events.
• 💱 Without replacement, the probability of subsequent picks changes as the number of desired outcomes and total outcomes change.
• 👾 Calculating the expected value helps determine if a game is profitable or not.

Questions & Answers

Q: What is the probability of picking the first green marble?

The probability of picking the first green marble is 3/5 or 60% since there are 3 green marbles out of a total of 5 marbles.

Q: Is the second pick's probability of being green the same as the first pick?

No, the second pick's probability depends on the first pick. If the first pick is green, the bag will have 4 marbles, and the probability of the second pick being green will be 2/4 or 50%.

Q: What is the total probability of picking two green marbles?

The total probability is the product of the probability of the first green (3/5) and the probability of the second green given the first is green (2/4). This gives us a total probability of 3/10 or 30%.

Q: Should you play the game based on the calculated probabilities?

No, it is not economically beneficial to play the game. The expected value is $0.30, and the cost to play is $0.35, resulting in a net loss.

Summary & Key Takeaways

• You are presented with a game where you need to pick two green marbles out of a bag with three green and two orange marbles.

• The probability of picking the first green marble is 3/5 (60%).

• The probability of picking two green marbles without replacement is calculated to be 3/10 (30%).

• It is concluded that playing the game is not economical since the expected value is $0.30 and the cost to play is $0.35.