Finding probabilities using binomial theorem

This article is a continuation from the previous binomial distribution article. It may be a good idea to go through it before attempting this section. This section explores how to use binomial theorem to find probabilities.

Example

A biased coin is to be thrown 10 times. Let p be the probability that the coin lands on heads when thrown. Find the probability of the coin landing heads;

1. 10 times
2. 6 times
3. 3 times

Let us find the probability of the coin landing heads 10 times when thrown 10 times.
We know the probability of getting heads is p. Here we assume that each throw is independent so we can simply multiply the probabilities at each throw together to obtain the probability when the thrown 10 times.
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Now we shall find the probability of landing heads when thrown 6 times;
When the coin does not land heads it must land tails. Here we shall let q = probability of landing tails. The coin can only land on heads or tails so we have a fixed number of probabilities. The number of arrangements with a total of 10 objects with 6 of which are unique can be written as the following. This is from the factorial notation <-[LINK] [IMAGE] ...so there is 210 number of arrangements of 6 heads and 4 tails. Therefore using the same idea as above p⁶ for the 6 heads, q⁴ for the 4 tails, so we have the expression as;
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Notice that q is (1 – p). The probability of getting tails is the probability of not getting heads and we know probabilities always add up to 1 and not less than 0. Therefore (1 – p) = q.
Lastly the probability of landing heads when thrown 3 times, that is;
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Notice that the probabilities above are terms in the following binomial expansion;
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You could use the full binomial expansion to write down the probability distribution for the random variable X.
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