David has two boxes containing shapes.
In box A there are 4 stars and 2 hearts.
In box B there are 2 stars and 1 heart.
David takes, at random, a shape from box A and puts it into box B.He then takes a shape from box B.
What is the probability that this shape is a star?
Ah, the probability question, a classic! Let's break it down, shall we?
In box A, there are a total of 6 shapes (4 stars + 2 hearts). Since David takes a shape from box A and puts it into box B, the total number of shapes in box B becomes 9 (6 shapes from A + 2 stars in B + 1 heart in B).
Now, the probability of taking a star from box B depends on the number of stars in box B. Previously, we had 2 stars in box B, and now we have an additional star transferred from box A. So, the total number of stars in box B is 3.
Therefore, the probability of getting a star from box B is 3/9, which simplifies to 1/3.
So, there you have it! The probability that the shape David takes from box B is a star is 1/3.
To find the probability that the shape David takes from box B is a star, we will calculate the likelihood of each step happening.
Step 1: David takes a shape from box A and puts it into box B. There are two possibilities for the shape he chooses: a star or a heart.
- Probability of choosing a star from box A: 4 stars out of 6 shapes = 4/6 = 2/3
- Probability of choosing a heart from box A: 2 hearts out of 6 shapes = 2/6 = 1/3
Step 2: David takes a shape from box B. After taking a shape from box A and putting it into box B, the total number of shapes in box B changes.
- Probability of choosing a star from box B:
- If David put a star in box B: 3 stars out of 7 shapes = 3/7
- If David put a heart in box B: 2 stars out of 7 shapes = 2/7
Now, we need to calculate the probability of each step happening consecutively.
Probability of choosing a star from box A AND then choosing a star from box B:
(2/3) * (3/7) = 6/21 = 2/7
Therefore, the probability that David takes a star from box B is 2/7.
To find the probability that the shape David takes from box B is a star, we need to consider the total number of shapes and the number of stars in box B after David transfers a shape from box A.
Let's break down the problem step by step:
Step 1: Calculate the number of stars in box A.
Box A contains 4 stars and 2 hearts, so there are a total of 4 + 2 = 6 shapes in box A.
Step 2: Calculate the number of stars in box B before David transfers a shape.
Box B initially contains 2 stars and 1 heart, so there are a total of 2 + 1 = 3 shapes in box B before any transfers.
Step 3: Calculate the probability of transferring a star from box A to box B.
The probability of transferring a star from box A to box B is calculated as the number of stars in box A divided by the total number of shapes in box A.
So, the probability of transferring a star from box A to box B is 4/6, which simplifies to 2/3 or approximately 0.667.
Step 4: Calculate the new number of stars in box B after the transfer.
Since David transfers a shape from box A to box B, the new number of stars in box B will be the original number of stars in box B (2) plus the number of stars transferred from box A (4/6 * 2 = 8/6 = 4/3).
So, the new number of stars in box B is 2 + 4/3 = 10/3 or approximately 3.333.
Step 5: Calculate the probability of picking a star from box B.
The probability of picking a star from box B is calculated as the number of stars in box B after the transfer divided by the total number of shapes in box B after the transfer.
So, the probability of picking a star from box B is (10/3) / (10/3 + 1) = 10/13 or approximately 0.769.
Therefore, the probability that the shape David takes from box B is a star is approximately 0.769 or 76.9%.
After the first transfer from A,
P(B=3stars, 1heart) = 2/3
P(B=2stars, 2hearts) = 1/3
So, the probability of getting a start when drawing from B is
2/3 * 3/4 + 1/3 * 1/2 = 2/3