Mathematics, 12.11.2020 19:10 isabellam646

A pair of dice is rolled 180 times. What is the probability that a total of 7 occurs
(a) at least 25 times?
(b) between 33 and 41 times inclusive?
(c) exactly 30 times?

Answers: 2

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Answer from: seonna

Using the normal approximation to the binomial, it is found that there is a:

a) 0.8643 = 86.43% probability that a total of 7 occurs at least 25 times.

b) 0.2978 = 29.78% probability that a total of 7 occurs between 33 and 41 times inclusive.

c) 0.0796 = 7.96% probability that a total of 7 occurs exactly 30 times.

In a normal distribution with mean $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.The binomial distribution is the probability of x successes on n trials, with p probability of a success on each trial. It can be approximated to the normal distribution with $\mu = np, \sigma = \sqrt{np(1-p)}$ .

In this problem:

When 2 dice are rolled, there are 36 possible outcomes, of which 6 of them[(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)] result in a sum of 7, thus the parameter p is $p = \frac{1}{6}$ .The pair of dice are rolled 180 times, thus n = 180

The mean and standard deviation are given by:

$\mu = np = 180\frac{1}{6} = 30$

$\sigma = \sqrt{np(1-p)} = \sqrt{180\frac{1}{6}\frac{5}{6}} = 5$

Item a:

Using continuity correction, this probability is $P(X \geq 25 - 0.5) = P(X \geq 24.5)$ , which is 1 subtracted by the p-value of Z when X = 24.5. Hence:

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{24.5 - 30}{5}$

Z = -1.1

Z = -1.1 has a p-value of 0.1357.

1 - 0.1357 = 0.8643.

0.8643 = 86.43% probability that a total of 7 occurs at least 25 times.

Item b:

Using continuity correction, this probability is $P(33 - 0.5 \leq X \leq 41 + 0.5) = P(32.5 \leq X \leq 41.5)$ , which is the p-value of Z when X = 41.5 subtracted by the p-value of Z when X = 32.5. Hence:

X = 41.5

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{41.5 - 30}{5}$

Z = 2.3

Z = 2.3 has a p-value of 0.9893.

X = 32.5

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{32.5 - 30}{5}$

Z = 0.5

Z = 0.5 has a p-value of 0.6915.

0.9893 - 0.6915 = 0.2978.

0.2978 = 29.78% probability that a total of 7 occurs between 33 and 41 times inclusive.

Item c:

Using continuity correction, this probability is $P(30 - 0.5 \leq X \leq 30 + 0.5) = P(29.5 \leq X \leq 30.5)$ , which is the p-value of Z when X = 30.5 subtracted by the p-value of Z when X = 29.5. Hence:

X = 30.5

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{30.5 - 30}{5}$

Z = 0.1

Z = 0.1 has a p-value of 0.5398.

X = 29.5

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{29.5 - 30}{5}$

Z = -0.1

Z = -0.1 has a p-value of 0.4602.

0.5398 - 0.4602 = 0.0796.

0.0796 = 7.96% probability that a total of 7 occurs exactly 30 times.

A similar problem is given at link

Answer from: cristinamiu8169

0.86433

0.29782

0.07966

Step-by-step explanation:

Given that :

Number of trials (n) = 180

Probability that a total of 7 occurs

To obtain a total of 7 from a pair of dice :

Combinations:

(1,6) ; (2,5) ; (3,4) ; (4,3) ; (5,2) ;(6, 1)

Sample space when a pair of dice is rolled = 6² = 36

Probability of success (p) = number of 7

combination possible ÷ sample space)

p = 6 / 36 = 0.1667

q = p' = 1 - p = 1 - 0.1667 = 0.8333

Using Normal approximation :

Mean (m) = np = 180 * 0.1667 = 30.006

Standard deviation (σ) = √npq

σ = √(180 * 0.1667 * 0.8333)

σ = 5.0003999

σ = 5.00

(a) at least 25 times?

P(x ≥ 25)

x = 25 - 0.5 = 24.5

Obtain the Zscore :

Z = (x - m) / σ

Z = (24.5 - 30) / 5

Z = -5.5/5

Z = -1.1

P(z ≤ - 1.1)

USing the z probability calculator :

P(z≥ - 1.1) = 0.86433

(b) between 33 and 41 times inclusive?

P(32.5 ≤ x ≤ 41.5)

Z =[ (32.5 - 30) / 5] ≤ x ≤ [ (41.5 - 30) / 5]

Z = 2.5/5 ≤ x ≤ 11.5 /5

Z = 0.5 ≤ x ≤ 2.3

Using z probability calculator :

P(z ≤ 0.5) = 0.69146

P(z ≤ 2.3) = 0.98928

0.98928 - 0.69146 = 0.29782

P(x = 30)

P(29.5 ≤ x ≤ 30.5)

Z =[ (29.5 - 30) / 5] ≤ x ≤ [ (30.5 - 30) / 5]

Z = - 0.5/5 ≤ x ≤ 0.5 /5

Z = - 0.1 ≤ x ≤ 0.1

Using z probability calculator :

P(z ≤ - 0.1) = 0.46017

P(z ≤ 0.1) = 0.53983

0.53983 - 0.46017 = 0.07966

Answer from: Quest

He na a

Answer from: Quest

You're doing something i dont know how to do but search up on google slope calculator maybe that will get you there
Choose the function that corresponds to each graph below.

Choose the function that corresponds to each graph below.

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A pair of dice is rolled 180 times. What is the probability that a total of 7 occurs
(a) at l...