MHB Legend Of ~Incredim's question at Yahoo Answers regarding binomial probability

[MarkFL]

Here is the question:

A coin is loaded so that the chance of getting heads in a single toss is 0.65. If the coin is tossed 400 times, estimate the probability?

of getting EXACTLY 275 heads

I have posted a link there to this thread so the OP can view my work.

 

[MarkFL]

Hello Legend Of ~Incredim,

First, we want to consider how many different ways there are to have 275 heads in a total of 400 tosses. This is equivalent to asking how may ways there are to choose 275 from 400, and is given by:

$${400 \choose 275}$$

Of these different choices, 275 are heads and have a probability of 0.65 and 125 are tails with a probability of 0.35. So, for one particular choice, for example the first 275 tosses are heads and the remainder are tails, we have by the special multiplication rule that the probability for that particular choice is:

$$(0.65)^{275}(0.35)^{125}$$

And then by the special addition rule, we find the total probability is:

$$P(\text{exactly 275 heads})={400 \choose 275}(0.65)^{275}(0.35)^{125}\approx0.0122157679732721$$

This result is what would be suggested by the binomial probability formula.