Find the probability that 200 tosses of a coin will result in (a) between 80 and 120 heads inclusive, (b) less than 90 heads, (c) less than 85 or more than 115 heads, (d) exactly 100 heads.

a)

b)  

c)

d)

Step-by-step explanation:

1) Notation and data given

n= 200 represent the total tosses

p(head)=p(tails)=1/2=0.5 if is a fair coin

The experiment on this case is tossing 200 times a coin

We can calculate np=200*0.5=100>10 and nq=n(1-p)=200*(1-0.5)=100>10

So then since np>10 and nq>10 we can use the approximation normal to the binomial distribution.

Let X our random variable who represents "the number of heads obtained in 200 tosses from a fair coin". This random variable X follows a normal distribution. And since we have all the conditions satisfied we can calculate the mean and the deviation for the normal distribution

Since X follows a normal distribution we can standarize on this way

And z is distributed normal with mean= and deviation =1.

This z score would be useful in order to calculate the probabilities required.

2) Part a

Using properties from the normal distribution we have this

3) Part b

And using a the normal standard distribution table or excel we find that:

4) Part c

Since the events and are independent, so we can find the probability like this

So we can find individually the probabilities like this:

So then:

5) Part d

If we use the normal approximation since the area below the curve for a point is not defined. Then the probability P(X=100) would be 0.

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step-by-step explanation:

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