# Help random variable

• Oct 10th 2007, 04:39 PM
ballinisahobby
Help random variable
The probability that a patient recovers from a stomach disease is .8. Suppose 20 people are known to have contracted this disease.

Let X = # of people who recover out of 20 contracted the disease
while PMF for X and solve problems. Assume that sample of size 20 is independently selected.

A. What is the probability that exactly 14 recover?
B. What is the probability that at least 10 recover?
C. What is the probability that at least 14 but not more than 18 recover?
D. What is the probability that at most 16 recover?
• Oct 10th 2007, 06:32 PM
Jhevon
Quote:

Originally Posted by ballinisahobby
The probability that a patient recovers from a stomach disease is .8. Suppose 20 people are known to have contracted this disease.

Let X = # of people who recover out of 20 contracted the disease
while PMF for X and solve problems. Assume that sample of size 20 is independently selected.

we can use Bernoulli trials here.

recall that by the method of Bernoulli Trials we have that, the probability of $k$ successes in $n$ trials is given by:

$P(k) = {n \choose k}p^kq^{n-k}$

where $p$ is the probability of success, and $q = 1 - p$ is the probability of failure.

Let recovery be a success here. thus p = 0.8, and q = 0.2 and n = 20

Quote:

A. What is the probability that exactly 14 recover?
this is $P(X = 14)$

Quote:

B. What is the probability that at least 10 recover?
this is $P(X \ge 10) = P(10) + P(11) + P(12) + ... + P(20)$

Quote:

C. What is the probability that at least 14 but not more than 18 recover?
this is $P(14 \le X \le 18) = P(14) + P(15) + ... + P(18)$

Quote:

D. What is the probability that at most 16 recover?
this is $P(X \le 16) = 1 - P(X > 16) = P(17) + P(18) + P(19) + P(20)$

this is a lot of computation, but you can reuse values, so it's not that bad