Probability using combinations | Probability and Statistics | Khan Academy

TL;DR

The probability of getting exactly 3 heads when flipping a fair coin 8 times is 7/32, or approximately 21.9%.

Transcript

So you might be wondering why I went off into permutations and combinations in the probability playlist, and I think you'll learn in this video. So let's say I want to figure out the probability-- I'm going to flip a coin eight times and it's a fair coin. And I want to figure out the probability of getting exactly 3 out of 8 heads. So I say 3/8 hea... Read More

Key Insights

• 🗂️ The probability of an event is the number of outcomes where the event occurs, divided by the total possible outcomes.
• 🤕 Choosing a specific number of flips resulting in heads can be represented using combinations.
• #️⃣ The formula for combinations, n choose k, helps determine the number of ways to choose k flips out of n.
• #️⃣ The intuition behind combinations involves considering the number of permutations and adjusting for the number of ways the chosen flips can be arranged.

Questions & Answers

Q: How do you calculate the probability of getting exactly 3 heads when flipping a coin 8 times?

The probability is determined by dividing the number of outcomes where exactly 3 heads are obtained (56) by the total possible outcomes (256). This gives a probability of 7/32 or approximately 21.9%.

Q: Can the formula for calculating combinations be used to solve this probability problem?

Yes, the formula for combinations, n choose k = n! / (k! * (n-k)!), can be used to calculate the number of ways to choose 3 flips out of 8. This can then be divided by the total number of outcomes to find the probability.

Q: What is the intuition behind the formula for calculating combinations?

The formula accounts for the number of ways to choose k items out of a total of n items, without considering the order. It takes into account the total number of permutations and divides by the number of ways the chosen items can be arranged.

Q: Can the same approach be used to calculate the probability of getting a different number of heads when flipping a coin?

Yes, the same approach can be used. Simply adjust the number of heads desired and the total number of flips accordingly. Use the formula for combinations to determine the number of ways to choose the desired number of flips resulting in heads.

Summary & Key Takeaways

• The probability is determined by the number of outcomes where exactly 3 heads are obtained, divided by the total possible outcomes.

• There are 2 to the power of 8 (256) possible outcomes when flipping a fair coin 8 times.

• Out of these outcomes, there are 56 ways to choose 3 flips that result in heads.