Conditional probability and combinations | Probability and Statistics | Khan Academy

TL;DR

Given a bag with 5 fair coins and 10 unfair coins, what is the probability of picking a fair coin if 4 out of 6 flips resulted in heads?

Transcript

Welcome back. Now let's do a problem that involves almost everything we've learned so far about probability and combinations and conditional probability. So let's say I have a bag again. And in that bag, I have 5 fair coins, and I have 10 unfair coins. And a fair coin, of course, is a 50:50 chance of getting heads or tails, and the unfair coin-- le... Read More

Key Insights

• 🤕 The probability of getting four out of six heads given a fair coin is 15/64.
• 🤕 The probability of getting four out of six heads given an unfair coin depends on the probability of heads and tails for the unfair coin.
• 🛻 The probability of picking a fair coin is 1/3, while the probability of picking an unfair coin is 2/3.
• 🤕 By applying Bayes' Theorem, the probability of picking a fair coin given four out of six heads is approximately 32.3%.

Questions & Answers

Q: What is the probability of picking a fair coin from the bag?

There are 5 fair coins out of a total of 15 coins, so the probability is 1/3 (or approximately 33.3%).

Q: What is the probability of getting four out of six heads, given a fair coin?

There are 15 possible combinations of getting four heads out of six flips, and each combination has a probability of (1/2)⁶. Therefore, the probability is 15/64 (or approximately 0.2344).

Q: What is the probability of getting four out of six heads, given an unfair coin?

For each combination of getting four heads out of six flips, the probability is (0.8)⁴(0.2)², as the unfair coin has an 80% chance of heads. There are 15 combinations, so the probability is 15(0.8)⁴(0.2)².

Q: What does Bayes' Theorem allow us to calculate in this problem?

Bayes' Theorem allows us to calculate the probability of picking a fair coin, given that we obtained four heads out of six flips.

Summary & Key Takeaways

• The problem involves a bag with 5 fair coins and 10 unfair coins, where the unfair coins have an 80% chance of heads and a 20% chance of tails.

• The goal is to find the probability of picking a fair coin, given that 4 out of 6 flips resulted in heads.

• Bayes' Theorem is used to calculate the probability, considering the probability of getting four heads given a fair coin, the probability of picking a fair coin, and the probability of getting four heads overall.