Find the probability fewer than half the villagers have the eye disorder? (binomial)

Question

In a certain african village, 80% of the villagers are known to have a particular eye disorder. 12 people are waiting to see the nurse.

$a)$ What is the most likely number to have the disorder?

$b)$ Find the probability that fewer than half have the eye disorder.

$c)$ Find the probability that 2 persons do not have the disorder.

Any help is appreciated!

Answer 1

By now, I'm guessing this is a problem from a book at a more elementary level than the Comments. (That's why we ask for 'context' as in one of the Comments. Without context it is not so easy to be helpful.)

Use the binomial formula to make a distribution table for the random variable $X \sim Binom(n = 12, p = .8),$ where $X$ is the number among 12 who have the disorder. Alternatively, depending on what text you are using, you may have such a table in an Appendix at the back of your text.

One example of using the binomial formula is $$P(X = 11) = {12 \choose 11}(.8)^{11}(.2) = 0.2061584.$$

Below is a table I made using statistical software. It has six-place accuracy. You may want to round your answers to four places.

   x  P( X = x )
   1    0.000000
   2    0.000004
   3    0.000058
   4    0.000519
   5    0.003322
   6    0.015502
   7    0.053150
   8    0.132876
   9    0.236223
  10    0.283468
  11    0.206158
  12    0.068719

(a) Find the biggest probability in the list. Answer is 10 patients.

(b) Half would be 6. Fewer than half would be 0, 1, 2, ..., or 5. Add probabilities for these six values of X. $P(X=0)$ is missing from the table because, as for $P(X=1),$ the probability is negligibly small. (Your sum should a little less than 0.04.)

(c) If exactly 2 do not, then exactly 10 do. What is the probability of that?

Now is the time to look in your text or lecture notes to see if anything I have said matches what is there, and how you might have worked out the answer on your own.