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

AI Thread Summary
The discussion focuses on calculating the probability of getting exactly 275 heads when tossing a loaded coin 400 times, with a head probability of 0.65. The number of ways to choose 275 heads from 400 tosses is represented by the binomial coefficient, calculated as 400 choose 275. The specific probability for one arrangement of 275 heads and 125 tails is determined using the multiplication rule, resulting in the expression (0.65)^275 * (0.35)^125. By applying the addition rule, the total probability is estimated to be approximately 0.0122, aligning with the binomial probability formula. This calculation provides a clear method for estimating outcomes in binomial scenarios.
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Here is the question:

Legend Of ~Incredim said:
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.
 
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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.
 
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