Math / Data & Statistics

QuestionAn urn contains 6 green and 8 pink balls. Five balls are randomly drawn from the urn in succession, with replacement. That is, after each draw, the selected ball is returned to the urn. What is the probability that all 5 balls drawn from the urn are green? Round your answer to three decimal places. (If necessary, consult a list of formulas.) $\square$

Studdy Solution

STEP 1

What is this asking? What are the chances of grabbing 5 green balls in a row from a mixed bag of green and pink balls, if we put the ball back every time? Watch out! Don't forget that we're putting the ball back each time – that makes a big difference!

STEP 2

1. Calculate the probability of drawing a green ball on a single draw.
2. Calculate the probability of drawing five green balls in a row.
3. Round the result.

STEP 3

We know there are 6 green balls and 8 pink balls.
So, the total number of balls is $6 + 8 = 14$ .

STEP 4

The probability of picking a green ball on any single draw is the number of green balls divided by the total number of balls.
This gives us $\frac{6}{14}$ , which simplifies to $\frac{3}{7}$ .
This is our single-draw green probability!

STEP 5

Since we're putting the ball back each time (that's what "with replacement" means), each draw is independent.
This means the result of one draw doesn't affect the others.

STEP 6

To get the probability of multiple independent events happening, we multiply their individual probabilities.
We want five green balls in a row, and the probability of a single green ball is $\frac{3}{7}$ .

STEP 7

So, the probability of five green balls in a row is $\frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} = \left(\frac{3}{7}\right)^5$ .

STEP 8

Calculating this gives us $\frac{3^5}{7^5} = \frac{243}{16807} \approx 0.014455$ .
This is the probability of five green balls in a row!

STEP 9

We're asked to round to three decimal places.
Our result, $0.014455$ , rounds to 0.014.

STEP 10

The probability of drawing five green balls in a row is approximately 0.014.

Was this helpful?

Studdy Solution

STEP 1

What is this asking? What are the chances of grabbing 5 green balls in a row from a mixed bag of green and pink balls, if we put the ball back every time? Watch out! Don't forget that we're putting the ball back each time – that makes a big difference!

STEP 2

1. Calculate the probability of drawing a green ball on a single draw.
2. Calculate the probability of drawing five green balls in a row.
3. Round the result.

STEP 3

We know there are 6 green balls and 8 pink balls.
So, the total number of balls is $6 + 8 = 14$ .

STEP 4

The probability of picking a green ball on any single draw is the number of green balls divided by the total number of balls.
This gives us $\frac{6}{14}$ , which simplifies to $\frac{3}{7}$ .
This is our single-draw green probability!

STEP 5

Since we're putting the ball back each time (that's what "with replacement" means), each draw is independent.
This means the result of one draw doesn't affect the others.

STEP 6

To get the probability of multiple independent events happening, we multiply their individual probabilities.
We want five green balls in a row, and the probability of a single green ball is $\frac{3}{7}$ .

STEP 7

So, the probability of five green balls in a row is $\frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} = \left(\frac{3}{7}\right)^5$ .

STEP 8

Calculating this gives us $\frac{3^5}{7^5} = \frac{243}{16807} \approx 0.014455$ .
This is the probability of five green balls in a row!

STEP 9

We're asked to round to three decimal places.
Our result, $0.014455$ , rounds to 0.014.

STEP 10

The probability of drawing five green balls in a row is approximately 0.014.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

What is this asking? What are the chances of grabbing 5 green balls in a row from a mixed bag of green and pink balls, if we put the ball back every time? Watch out! Don't forget that we're putting the ball back each time – that makes a big difference!

STEP 2

1. Calculate the probability of drawing a green ball on a single draw.2. Calculate the probability of drawing five green balls in a row.3. Round the result.

STEP 3

We know there are **6** green balls and **8** pink balls.So, the **total** number of balls is 6+8=146 + 8 = 146+8=14.

STEP 4

The probability of picking a green ball on any single draw is the number of green balls divided by the total number of balls.This gives us 614\frac{6}{14}146​, which simplifies to 37\frac{3}{7}73​.This is our **single-draw green probability**!

STEP 5

Since we're putting the ball back each time (that's what "with replacement" means), each draw is **independent**.This means the result of one draw doesn't affect the others.

STEP 6

To get the probability of multiple independent events happening, we **multiply** their individual probabilities.We want five green balls in a row, and the probability of a single green ball is 37\frac{3}{7}73​.

STEP 7

So, the probability of five green balls in a row is 37⋅37⋅37⋅37⋅37=(37)5\frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} = \left(\frac{3}{7}\right)^573​⋅73​⋅73​⋅73​⋅73​=(73​)5.

STEP 8

Calculating this gives us 3575=24316807≈0.014455\frac{3^5}{7^5} = \frac{243}{16807} \approx 0.0144557535​=16807243​≈0.014455.This is the **probability of five green balls in a row**!

STEP 9

We're asked to round to three decimal places.Our result, 0.0144550.0144550.014455, rounds to **0.014**.

STEP 10

The probability of drawing five green balls in a row is approximately **0.014**.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Calculate the probability of drawing a green ball on a single draw.
2. Calculate the probability of drawing five green balls in a row.
3. Round the result.

We know there are 6 green balls and 8 pink balls.
So, the total number of balls is $6 + 8 = 14$ .

The probability of picking a green ball on any single draw is the number of green balls divided by the total number of balls.
This gives us $\frac{6}{14}$ , which simplifies to $\frac{3}{7}$ .
This is our single-draw green probability!

Since we're putting the ball back each time (that's what "with replacement" means), each draw is independent.
This means the result of one draw doesn't affect the others.

To get the probability of multiple independent events happening, we multiply their individual probabilities.
We want five green balls in a row, and the probability of a single green ball is $\frac{3}{7}$ .

So, the probability of five green balls in a row is $\frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} \cdot \frac{3}{7} = \left(\frac{3}{7}\right)^5$ .

Calculating this gives us $\frac{3^5}{7^5} = \frac{243}{16807} \approx 0.014455$ .
This is the probability of five green balls in a row!

We're asked to round to three decimal places.
Our result, $0.014455$ , rounds to 0.014.

The probability of drawing five green balls in a row is approximately 0.014.