Integrated math 2
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.
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- 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?(22 votes)
- 2 outcomes for one event; 3 events. Therefore 2^3 = 8 outcomes.(50 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
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.(18 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.(0 votes)
- 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(5 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)
- if from a 100 sided polygon 3 points are chosen at random. Then what is the probability that the chosen points form a right angled triangle? I know the answer is 1/33 but I can't solve it.(2 votes)
- I might be wrong about this, but this is how I would prove it:
Let's start with the first point. That point could form 100 different angles with the next point and the horizontal axis: approximately 0, 3.6, 7.2, 10.8.......up to 180. The reason that they are all multiples of 3.6 is because 3.6 is approximately equal to 360(a full circle of degrees) divided by 99(the number of points on the polygon).
So the first point could be anywhere. The second point has to form a complementary angle with the first one, the x-axis, and the third point to make a right angle. Example: if point A forms an angle of about 10.8 with point B and the x-axis, then point B has to make the angle 349.2 with point C and the x-axis. The probability of point B making that exact angle is 1/99, since it has to connect with point C, which could be randomly at any of the 99 points. So the probability would be 1/99, except that we have to remember that this layout could start with any point, not just point A. It could start with point B or point C. That triples it's chances of happening, so the final probability would be
Hopefully this was clear enough. A diagram would be clearer, but I can't write one here. :)(2 votes)
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.