If a fair die is rolled 6 times, what is the, №13500541, 02.12.2022 18:52

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If a fair die is rolled 6 times, what is the probability, rounded to the nearest thousandth, of getting at least 5 sixes?

Step-by-step answer

P Answered by Studen AI

1

To solve this task, we can use the concept of binomial distribution. Let's break down the problem step by step using math laws and formulas:

Step 1: Determine the number of possible outcomes per roll of the fair die.
A fair die has six sides, numbered from 1 to 6. Therefore, the number of possible outcomes per roll is 6.

Step 2: Determine the probability of getting a six in a single roll.
Since the die is fair, each of the six outcomes has an equal probability of occurring. The probability of rolling a six is given by 1/6.

Step 3: Determine the probability of getting at least 5 sixes in 6 rolls.
To find the probability of getting at least 5 sixes, we need to consider all possible combinations: getting exactly 5 sixes, getting exactly 6 sixes, or any combination in between.

The formula to calculate the probability of getting exactly k successes in n independent trials is given by the binomial coefficient formula:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

Where:
- n is the total number of trials (6 rolls)
- k is the number of successful outcomes (5 or 6 sixes)
- p is the probability of a single successful outcome (1/6)

Step 4: Calculate the probability of getting exactly 5 sixes.
Using the binomial coefficient formula, we have:

P(X = 5) = (6 choose 5) * (1/6)^5 * (5/6)^(6 - 5)

Calculating this expression:
(6 choose 5) = 6! / (5!(6-5)!) = 6
(1/6)^5 = 1/7776
(5/6)^(6-5) = 5/6

P(X = 5) = 6 * (1/7776) * (5/6) = 30/46656

Step 5: Calculate the probability of getting exactly 6 sixes.
Using the binomial coefficient formula, we have:

P(X = 6) = (6 choose 6) * (1/6)^6 * (5/6)^(6 - 6)

Calculating this expression:
(6 choose 6) = 1
(1/6)^6 = 1/46656
(5/6)^(6-6) = 1

P(X = 6) = 1 * (1/46656) * 1 = 1/46656

Step 6: Calculate the probability of getting at least 5 sixes.
To find the probability of getting at least 5 sixes, we need to sum up the probabilities of getting exactly 5 sixes and exactly 6 sixes:

P(X >= 5) = P(X = 5) + P(X = 6) = 30/46656 + 1/46656 = 31/46656

Step 7: Round the probability to the nearest thousandth.
The probability, 31/46656, rounded to the nearest thousandth is approximately 0.001.

Therefore, the probability, rounded to the nearest thousandth, of getting at least 5 sixes when rolling a fair die 6 times is 0.001.

Please note that I have checked my work at each step and ensured that no terms are omitted.

Mathematics

If a standard 6 sided dice is rolled. (Write your answers as a decimal, rounded to the nearest hundredth.) Find: a. Probability of rolling one die and it landing on both a prime number and a multiple of two. b.Probability of rolling one die and it landing on a 3 or a prime number. c. P(3 U Multiple of 2) = d. P(2 U Even number) =

Step-by-step answer

P Answered by PhD

1

see below

Step-by-step explanation:

The possible outcomes are 1,2,3,4,5,6

P( a prime number and a multiple of two)

The only prime number that is a multiple of 2 is 2

P( a prime number and a multiple of two) = number of outcomes / total outcomes

P( a prime number and a multiple of two) = 1/6 = .17

P(3 or a prime number)

The prime numbers are 2,3,5 and 3 is included in this list so we have 3 outcomes

P( 3 or a prime number) = number of outcomes / total outcomes

P( 3 or a prime number) =3/6 =1/2= .5

P(3 U Multiple of 2)

Multiples of 2 are 2,4,6 and add 3 to the list so there are 4 outcomes

P( 3 U Multiple of 2) = number of outcomes / total outcomes

P( 3 or a prime number) =4/6 =2/3= .67

P(2 U Even number)

even numbers are 2,4,6 and 2 is on the list so there are 3 outcomes

P( 2 U Even number) = number of outcomes / total outcomes

P( 3 or a prime number) =3/6 =1/2= .5

Mathematics

If a standard 6 sided dice is rolled. (Write your answers as a decimal, rounded to the nearest hundredth.) Find: a. Probability of rolling one die and it landing on both a prime number and a multiple of two. b.Probability of rolling one die and it landing on a 3 or a prime number. c. P(3 U Multiple of 2) = d. P(2 U Even number) =

Step-by-step answer

P Answered by PhD

1

see below

Step-by-step explanation:

The possible outcomes are 1,2,3,4,5,6

P( a prime number and a multiple of two)

The only prime number that is a multiple of 2 is 2

P( a prime number and a multiple of two) = number of outcomes / total outcomes

P( a prime number and a multiple of two) = 1/6 = .17

P(3 or a prime number)

The prime numbers are 2,3,5 and 3 is included in this list so we have 3 outcomes

P( 3 or a prime number) = number of outcomes / total outcomes

P( 3 or a prime number) =3/6 =1/2= .5

P(3 U Multiple of 2)

Multiples of 2 are 2,4,6 and add 3 to the list so there are 4 outcomes

P( 3 U Multiple of 2) = number of outcomes / total outcomes

P( 3 or a prime number) =4/6 =2/3= .67

P(2 U Even number)

even numbers are 2,4,6 and 2 is on the list so there are 3 outcomes

P( 2 U Even number) = number of outcomes / total outcomes

P( 3 or a prime number) =3/6 =1/2= .5

Mathematics

Plato a six-sided die of unknown bias is rolled 20 times, and the number 3 comes up 6 times. in the next three rounds (the die is rolled 20 times in each round), the number 3 comes up 6 times, 5 times, and 7 times. the experimental probability of rolling a 3 is %, which is approximately % more than its theoretical probability. (round off your answers to the nearest integer.)

Step-by-step answer

P Answered by Specialist

59

The experimental probability of rolling a 3 is 30%, which is approximately 13% more than its theoretical probability.

Mathematics

Asix-sided die of unknown bias is rolled 20 times, and the number 3 comes up 6 times. in the next three rounds (the die is rolled 20 times in each round), the number 3 comes up 6 times, 5 times, and 7 times. the experimental probability of rolling a 3 is %, which is approximately % more than its theoretical probability. (round off your answers to the nearest integer.)

Step-by-step answer

P Answered by PhD

152

Out of 80 rolls, the number three has come up 6+6+5+7=24 times. As a result, we can estimate the probability as 24/80=3/10, or 30%. The probability of rolling a three on a standard die is 16.7%, so the difference is 30%-16.7%=13.3%, or approximately 13%.

Mathematics

Type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s). a six-sided die of unknown bias is rolled 20 times, and the number 3 comes up 6 times. in the next three rounds (the die is rolled 20 times in each round), the number 3 comes up 6 times, 5 times, and 7 times. the experimental probability of rolling a 3 is %, which is approximately % more than its theoretical probability. (round off your answers to the nearest integer.)

Step-by-step answer

P Answered by PhD

51

The experimental probability of getting a 3 on a die is 30% which is approximately 13% more than its theoretical probability (17%).

Step-by-step explanation:

Theoretical probability of number 3 on a die:

Total no. of possibilities = 6

Probability of getting a 3 on a die each time it is rolled = 1/6

= 0.16667

= 17%

Experimental probability of number 3 on a die:

Total no. of rounds = 4

Rolls each round = 20

no. of 3s in round 1 = 6

no. of 3s in round 2 = 6

no. of 3s in round 3 = 5

no. of 3s in round 4 = 7

Total rolls = 20*4 = 80

no. of times 3 comes up = 6+6+5+7 = 24

Experimental probability of getting a 3 on a die

each time it is rolled = no. of 3s/total rolls

= 24/80

= 0.3

= 30%

Difference = 30% - 17% = 13%

Experimental probability of getting a 3 on a die is 30% which is approximately 13% more than its theoretical probability (17%).

Mathematics

Plato a six-sided die of unknown bias is rolled 20 times, and the number 3 comes up 6 times. in the next three rounds (the die is rolled 20 times in each round), the number 3 comes up 6 times, 5 times, and 7 times. the experimental probability of rolling a 3 is %, which is approximately % more than its theoretical probability. (round off your answers to the nearest integer.)

Step-by-step answer

P Answered by Master

59

The experimental probability of rolling a 3 is 30%, which is approximately 13% more than its theoretical probability.

Mathematics

Type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s). a six-sided die of unknown bias is rolled 20 times, and the number 3 comes up 6 times. in the next three rounds (the die is rolled 20 times in each round), the number 3 comes up 6 times, 5 times, and 7 times. the experimental probability of rolling a 3 is %, which is approximately % more than its theoretical probability. (round off your answers to the nearest integer.)

Step-by-step answer

P Answered by PhD

51

The experimental probability of getting a 3 on a die is 30% which is approximately 13% more than its theoretical probability (17%).

Step-by-step explanation:

Theoretical probability of number 3 on a die:

Total no. of possibilities = 6

Probability of getting a 3 on a die each time it is rolled = 1/6

= 0.16667

= 17%

Experimental probability of number 3 on a die:

Total no. of rounds = 4

Rolls each round = 20

no. of 3s in round 1 = 6

no. of 3s in round 2 = 6

no. of 3s in round 3 = 5

no. of 3s in round 4 = 7

Total rolls = 20*4 = 80

no. of times 3 comes up = 6+6+5+7 = 24

Experimental probability of getting a 3 on a die

each time it is rolled = no. of 3s/total rolls

= 24/80

= 0.3

= 30%

Difference = 30% - 17% = 13%

Experimental probability of getting a 3 on a die is 30% which is approximately 13% more than its theoretical probability (17%).

Mathematics

Find the probability that you roll an odd number on a die and then spin a spinner with numbered 1 to 6 and get an even number. enter your answer as a decimal, rounded to the nearest hundredth.

Step-by-step answer

P Answered by PhD

2

One half the numbers on a die are odd, (1,3,5) so the probability of that would be 1/2.

One half of the numbers on the spinner are even (2,4,6) so the probability of that would also be 1/2.

The probability of both happening would be 1/2 x 1/2 = 1/4 probability.

Mathematics

Find the probability that you roll an odd number on a die and then spin a spinner with numbered 1 to 6 and get an even number. enter your answer as a decimal, rounded to the nearest hundredth.

Step-by-step answer

P Answered by PhD

2

One half the numbers on a die are odd, (1,3,5) so the probability of that would be 1/2.

One half of the numbers on the spinner are even (2,4,6) so the probability of that would also be 1/2.

The probability of both happening would be 1/2 x 1/2 = 1/4 probability.

If a fair die is rolled 6 times, what is the probability, rounded to the nearest thousandth , of getting at most sixes?

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