CDF of

BinomialDistribution is an example of a right-continuous function:

A basketball player has a free-throw percentage of 0.75. Simulate 10 free throws:

Find the probability that the player hits 2 out of 3 free throws in a game:

Find the probability that the player hits the 2 last of 5 free throws:

Find the expected number of hits in a game with n free throws:

A baseball player is a 0.300 hitter. Simulate 5 pitches:

Find the expected number of hits if the player comes to bat 3 times:

A drug has proven to be effective in 30% of cases. Find the probability it is effective in 3 of 4 patients:

Find the expected number of successes when applied to 500 cases:

The number of heads in

n flips with a fair coin can be modeled with

BinomialDistribution:

Show the distribution of heads for 100 coin flips:

Compute the probability that there are between 60 and 80 heads in 100 coin flips:

Now, suppose that for an unfair coin the probability of heads is 0.6:

The distribution and the corresponding probabilities have changed:

A machine produces parts, with 1 in 10 defective:

Compute the probability that at most 1 of 5 parts is defective:

An airplane engine fails with probability p; compute the probability that no more than 2 of 4 fail:

Compute the probability that no more than 1 of 2 fails:

Decide when the choice of four engines is better than two engines:

A system uses triple redundancy with three microprocessors and is designed to operate as long as one processor is still functional. The probability that a microprocessor is still functional after

seconds is

. Find the probability that the system is still operating after

seconds:

With mean time to failure for each processor

, find out when the system functions with a probability of less than 99%:

Expressed in years:

Gary Kasparov, chess champion, plays in a tournament simultaneously against 100 amateurs. It has been estimated that he loses about 1% of such games. Find the probability of losing 0, 2, 5, and 10 games:

Use a Poisson approximation to compute the same probabilities:

Perform the same computation when he is playing 5 games, but with stronger opposition so that his loss probability is 10% instead:

In this case the Poisson approximation is less accurate:

A packet consisting of a string of

n symbols is transmitted over a noisy channel. Each symbol has probability

of incorrect transmission. Find

n such that the probability of incorrect packet transmission is less than

:

Compute the same limit using a Poisson approximation:

Find the probability that

out of

n customers need a service if each uses it with probability

p:

Compute the probability that more than

(capacity) simultaneous service requests are made:

Compute the probability of getting service if

and

for different capacities

:

Find the smallest capacity

that provides a 99.9% probability of getting service:

Two players roll dice. If the total of both numbers is less than 10, the second player is paid 4 cents; otherwise the first player is paid 9 cents. Is the game fair?:

The game is not fair, since mean scores per game are not equal:

Find the probability that after n games the player at the disadvantage scores more:

The probability exhibits oscillations:

The maximum of probability is attained at

: