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# Binomial distribution

Sal introduces the binomial distribution with an example.

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• I'm confused with these factorials (!) In the problem around how does 5!/4!=5? •  ``5! = 5•4•3•2•14! = 4•3•2•15!   5•4•3•2•1── = ─────────4!     4•3•2•15!   ── = 54!``
• I believe Sal's approach of using 5Cx/2^5 where x is the exact number of heads only works because we have assumed a fair coin here. How would we solve this problem if, say the probability of heads on our coin was 60%? I think we would have to use something involving Bernoulli trials. • Why are probabilitiy distributions so often simetrical? • If 15 dates are selected at random, what is probability of getting two Sundays? • I have a very silly question. Ya I know it's would be very silly to ask but I can't resist asking this! How can you know that for getting a consecutive 2 or 3 or 4 heads in a row, you will multiply the probability of each event.
I know that sum them up would be nonsense cause for example with 3 consecutive heads: 0.5+0.5+0.5 = 1.5 > 1.
Still I want to intuitively feel why multiplication is the best here?? Tks a lot! • Have you looked at the tree idea before? Say the tree has a stem of 1 which we split three ways so every one of three branches has 1/3 "thickness". Now if you grow three branches out of that one, they add up to 1 * "thickness" of the parent branch. But that one isn't 1, it's 1/3 of the stem thickness. The daughter branches are each 1/3 "thickness" of the parent for example. But that must mean they are 1/3 * 1/3 * stem which I arbitrarily chose to be 1. Does that image make a little bit more sense? It's not limited to binomial distribution problems as I tried to indicate with the three branches instead of 2 (binary).
• If we have to flip a coin 10 times, what is the probability of getting tails 7 times? • Does addition of all the values give the total probability? • It depends on what you mean by total probability. The probability of having exactly 2 heads is 5/32, which is 15.625% or 0.15625
To arrive at that probability, we didn't do any addition.

If you mean using addition so we can find the total possible number of ways that the coins can be flipped, it is true that we can add the number of ways that there can be zero head, plus the number of ways there can be one head, plus the number of ways there can be 2 heads, plus the number of ways there can be 3 heads, plus the number of ways there can be 4 heads, plus the number of ways there can be 5 heads.
1 + 5 + 10 + 10 + 5 + 1 = 32 different outcomes when you flip a coin 5 times
If you add the fractions, you get
1/32 + 5/32 + 10/32 + 10/32 + 5/32 + 1/32 = 32/32 = 1/1 = 1
The answer of one doesn't tell you much about the coin flip outcomes, unless you are checking that the probability of zero heads plus the probability of one head plus the probability of two heads plus the probability of three heads plus the probability of four heads plus the probability of five heads will add up to 100 percent of the total outcomes.
In other words, the probability that you will get either 0, 1, 2, 3, 4, or 5 heads is 1 (which is 100%)
• I came across a question which asked me what is THE EXPANSION OF THE BIONOMIAL in algebra. What does that mean? • To make sure I get it right "6_C_3" is the notation of 6! / [ 3! (6-3)! ], whis is interpreted as out of 6 different combination, you are interested of the of the probability of getting 3 you are intereste in. Is that correct? But we decide to sketch the tree, that is branching off in two different directions or even more? I am a little confused. For example if we had an imaginary coin with three sides, how would be the notation?  