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### Course: Precalculus (2018 edition) > Unit 6

Lesson 8: Probability using combinatorics- Probability using combinations
- Probability & combinations (2 of 2)
- Probability with counting outcomes
- Getting exactly two heads (combinatorics)
- Exactly three heads in five flips
- Generalizing with binomial coefficients (bit advanced)
- Example: Different ways to pick officers
- Example: Combinatorics and probability
- Example: Lottery probability
- Probability with permutations and combinations
- Mega millions jackpot probability
- Birthday probability problem

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# Probability with counting outcomes

The probability of getting exactly 2 heads when flipping three coins. Thinking about this by visualy depicting all of the outcomes. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- At0:15, Sal begins to list all the possible outcomes for flipping 3 coins. Is there any way to figure out the number of outcomes without listing them or using a chart?(29 votes)
- 2 outcomes for one event; 3 events. Therefore 2^3 = 8 outcomes.(64 votes)

- How many people are needed in a room so that the probability of two people sharing the same birthday is roughly one-half? I know the answer is 23, but I do not know how.(15 votes)
- a: Possible birthday permutations: 365^n

b: Permutations with no shared birthdays: 365! / (365-n)

Probability of n people NOT sharing birthday:

b / a = 365! / (365-n)! / 365^n = 365! / (365-n)! / 365^n

n=23:

365! / (365-23)! / 365^23 = 0.493

Probability of one or more shared birthday between 23 people:

1 - b/a = 0.507

I*think*that's an acceptable way to do it, but I've been wrong before.(20 votes)

- What is the difference between theoretical probability and experimental probability?(10 votes)
- theoretical probability says it's 99% will happen, but i can also get the 1% in the experimental probability.(1 vote)

- What if we change this to 5 coins?(4 votes)
- See Probability is nothing but
**Expected Events / All Possible Events**

So here if we flip 5 coins then all possible events =**2 * 2 * 2 * 2 * 2**=**2^5**=**32**

Expected event is getting exactly two head, if you know how combination works then you may find that event of getting exact 2 heads in 5 flips is nothing but**5C2**=**10**

So Probability of getting exactly two heads in 5 flips =**10/32**OR**5/16**(6 votes)

- How am i supposed to solve this with a tree diagram?(7 votes)
- draw t, h in the first row, each of those branch out into t and h, so there will be 2 ts and 2hs in the second row, then those branch out into 2 ts and hs each, so there are 4ts and 4hs in the third row(2 votes)

- Thank you for the awesome lecture :) I suddenly got a question on my mind. I guess it's about probability and possiblity? Okay. Let's say we have to find a genious student. We had total 100 students, but couldn't find the genious one. Than if there are more than 100 students, like 500 or something? Wouldn't the possibility to find the genious student can be higher? But I think this theory contradicts with the probabilitical method. Whoa I'm getting more confused as I write it.(6 votes)
- You are on the right track. Given that there are genius students with in the population, the larger the samples size of the population you take, the greater the possibility (that is to say, it is more probable it is) that your sample will have a genius student with in it.

So, with a smaller sample size, eg 100 students, it is less likely you will find a genius student within it than if your sample size was 500 students, but that does not mean the finding a genius student in the smaller size is impossible, it could happen, but is just less likely to happen.

Keep it up!

Keep thinking and asking questions!(4 votes)

- Why is THH and HHT considered two different possible outcomes? Are they not the same outcome?(5 votes)
- When 3 coins (or any other specified number of coins) are tossed, every
**sequence**of heads and/or tails is equally likely. Each possible sequence occurs with probability (1/2)(1/2)(1/2) = 1/8 (or more generally with probability 1/(2^n) if n coins are tossed). However, not every combination of heads and/or tails without regard to order is equally likely. So in this type of situation, order of heads and/or tails is important for computing probabilities.

Have a blessed, wonderful day!(5 votes)

- So I get that a deck of cards may be 52!. But, as I play some other card games I was wondering how I'd work out the possible hand combinations if say my deck contains duplicate cards?:

( do I just minus them from n? ) i.e (n-**minus duplicates**)!/*k_!*1/_k*?(3 votes)- If you have a deck of n cards, there are n! ways of arranging them.

Now if you want to figure out how many different hands you could get with 5 cards in hand, using a deck of 52 cards for instance, you'd use the "choose function" Which will come soon after in the probability section, but here is a "sneak peak". the choose function takes 2 parameters, n = how many things to choose from (here 52, all the cards) and how many do we want to pick (here 5 for our example)

choose(52,5) = 52! / (52-5)! * 5! = 2,598,960

To generalize it, let's say N is total of cards to choose from, and K the number of cards you'll have in your hand

choose(n,k)= n! / (n-k)! * k!(4 votes)

- Would this be a permutation or a combination?(4 votes)
- Permutation, I believe.(1 vote)

- Wouldn't this be 1/3 because there are only two options on each coin? There are not 8 options because there are multiples of each action but just in different orders.(3 votes)
- The total number of outcomes is indeed 2^3 = 8, as each coin has 2 options (heads or tails), and there are 3 coins. However, not all outcomes are equally likely. To find the probability of exactly 2 heads, you need to consider the number of favorable outcomes (those with exactly 2 heads) out of the total.(1 vote)

## Video transcript

Find the probability
of flipping exactly two heads on 3 coins. So to figure out this
probability, a good place to start is just to think about all
of the different possible ways that we can flip 3 coins. So we could get all tails. Tails, tails, tails. We could get tails,
tails, heads. We could get tails,
heads, tails. We could get tails,
heads, heads. We could get heads,
tails, tails. We could get heads,
tails, heads. We could get heads,
heads, tails. And then we could
get all heads. We could get all heads
over here. So there are 1, 2, 3, 4, 5,
6, 7, 8 possible outcomes. 8 possible outcomes. Now how many of the
outcomes involve flipping exactly 2 heads? Let's see, that's all tails. That's 1 head, 1 head. This has 2 heads right there. That's 1 head. This is 2 heads right
over there. Then this is 2 heads
right over here. And then this is 3 heads,
so that doesn't count. So there are 3 outcomes
with exactly 2 heads. So, let me spell
heads properly. 2 heads. So the probability of flipping
exactly 2 heads-- And the word exactly is important, because
if you didn't say exactly, then maybe 3 heads, when you
flip 2 heads, so we have to say exactly 2 heads. So you don't include
the situation where you get 3 heads. So the probability of flipping
exactly 2 heads is equal to the 3 outcomes with 2 heads
divided by the 8 possible outcomes, or 3/8. So it is equal to 3/8. And we are done.