Question

If I toss a fair coin 50,000 times which of the following is true?

a) the number of heads should be between 15,000 and 25,000.

b) the proportion of heads should be close to 50%.

c) the proportion of heads in these tosses is a parameter.

d) the number of heads should be exactly 25,000.

e) the proportion of heads will be close to 1.

Answer 1

When we flip a coin, there are two possible outcomes: heads and tails. Each outcome has a fixed probability, the same from trial to trial. In the case of coins, heads and tails each have the same probability of 1/2.

Here the coin is tossed 50000 times and thus number of trials ​ is 50000. This experiment has only two outcomes ​ which has equal probability of occurence. Hence it is a binomial random experiment.

Now if we define head as a success, the probability of occurence is . Thus the proportion of heads is close to 50%.

TO FIND NUMBER OF HEADS:

Since tossing a coin 50000 times follows binomial distribution where the probability of success (obatining heads) is ​.

The 95% confidence interval within which mean number of heads lie is calculated using the formula,

Mean number of heads​ (2*Standard error)

MEAN NUMBER OF HEADS:

Now to find expected (mean) number of heads, we use the mean of binomial random experiment .

Thus number of heads in this random experiment is

The standard deviation of binomial distribution (SD)

  

Now the standard error is given by,

  

  

  

CONFIDENCE INTERVAL:

The 95% confidence interval within which mean number of heads lie is calculated using the formula,

Mean number of heads​ (2*Standard error)

There is not much difference between lower and upper 95% confidence limits; both are 25000, thus the number of heads should be exactly 25,000.

Since tossing a coin 50000 times is a random experimet with two outcomes ​ which has equal probability of occurence. Thus proportion of heads will not be close to 1 but close to 50%. And we have calculated expected number of heads to be 25000.

Thus the correct options (true statements) are Option (b) The proportion of heads should be close to 50% and Option (d) The number of heads should be exactly 25,000.