# Binomial distribution

### Binomial distribution

#### Lessons

$P(x)={_n}C_x \;P^x(1-p)^{n-x}$

$n$: number of trials
$x$: number of success in n trials
$p$: probability of success in each trial
$P(x)$: probability of getting $x$ successes (out of $n$ trials)

$\cdot$ binomialpdf $(n,p,x)$
$\cdot$ binomialcdf $(n,p,x)$
• Introduction
a)
Binomial

b)
Binomial Formula

c)
Binomialpdf Calculator

• 1.
Identify which of the following experiments below are binomial distributions?

i. A fair die is rolled 4 times. What is the probability of the one coming up 2 times?

ii. A fair coin is flipped until head comes up 7 times. What is the probability that the coin will be flipped 10 times?

iii. 1,000,000 nails are produced in a factory a day. If each nail has a probability of 0.5% of being defective (something being wrong with that nail), then what is the probability that less than 50 nails will be defective in a day?

iv. Roughly 7.5% of Canadians have some form of heart disease. If 100 Canadians are sampled what is the probability that 10 of them will have heart disease?

v. If 5 cards are drawn from a deck, what is the probability that 2 of them will be hearts?

vi. If a fair die is rolled 8 times, what is the probability of getting 2 fours and 3 sixes?

• 2.
An urn contains 6 red balls and 4 green balls. A total of 5 balls are drawn; list all the different combinations of red balls that can be drawn in each of the following cases:

i. A total of 3 green balls are drawn

ii. At most 3 red balls are drawn

iii. At least 2 red balls are drawn

iv. Less than 4 red balls are drawn

v. More than 3 green balls are drawn

• 3.
A die is rolled 3 times, what is the probability that a four is rolled exactly 2 times?

• 4.
A coin is flipped 20 times, what is the probability that the coin comes up heads 15 times?

• 5.
Jimmy the Joker is an unfair gambler. He weights a die so it rolls a "6" with 75% chance. He then bets that if he rolls his die 4 times he will roll six exactly 3 times. What is his probability of winning this bet?

• 6.
Thomas is packing for a trip and wants to bring some stuffed animals along for comfort. He owns 8 stuffed animals, and will pack each stuffed animals independently of all the others with a probability of 30%. Determine the probability that he takes;
a)
0 stuffed animals along with him.

b)
1 stuffed animal with him

c)
at most two animals along with him.

d)
at most 5 animals along with him.

e)
at least 6 animals along with him.