Suppose that a ball is drawn at random from the box (a), its color is recorded, and then the ball is put back in the box. Draw a tree diagram for this experiment and find the probability that the two balls are of different colors.
Assuming the probability is that a child is a boy and births are independent. What is the probability that if a family will have four children (no twins), they will all be boys?
To draw a tree diagram for this experiment, we need to consider the two possible outcomes for each ball: the ball can be either red or blue. Let's represent red as R and blue as B.
The first step is to draw the branches representing the possible outcomes for the first ball:
R
/
/
/
/
a
\
\
\
B
At each branch, we draw two more branches representing the possible outcomes for the second ball, which are also red or blue:
R R
/ /
/ /
/ /
/ /
a a
\ \
\ \
\ \
B B
To find the probability that the two balls are of different colors, we need to identify the outcomes where the first ball is red and the second ball is blue, or vice versa.
These outcomes are represented by the branches highlighted in green below:
R (1/2) R (1/2)
/ /
/ /
/ /
/ /
a a
\ \
\ \
\ \
B (1/2) B (1/2)
Since each branch in the tree diagram represents an equally likely outcome, we can find the probability of getting two balls of different colors by adding the probabilities of the highlighted branches:
P(two balls of different colors) = P(RB or BR) = P(R) * P(B|R) + P(B) * P(R|B)
P(R) = 1/2 (probability of selecting a red ball)
P(B|R) = 1/2 (probability of selecting a blue ball given that the first ball is red)
P(B) = 1/2 (probability of selecting a blue ball)
P(R|B) = 1/2 (probability of selecting a red ball given that the first ball is blue)
Plugging in these values, we get:
P(two balls of different colors) = (1/2) * (1/2) + (1/2) * (1/2) = 1/4 + 1/4 = 1/2
Therefore, the probability that the two balls are of different colors is 1/2 or 0.5.
To draw a tree diagram for this experiment, follow these steps:
1. Start by drawing a branch for the first ball being drawn. Label this branch with the possible outcomes, which in this case are the colors of the ball (let's say red, blue, and green).
2. From each outcome in the first branch, draw another branch for the second ball being drawn. Label these branches with the possible outcomes, which are again the colors of the ball (red, blue, and green).
3. Repeat step 2 for each outcome in the first branch.
The resulting tree diagram should have three branches at the top representing the three possible outcomes for the first ball, and three branches extending from each of these three branches, representing the three possible outcomes for the second ball.
To find the probability that the two balls are of different colors, count the number of branches where the two balls have different colors and divide it by the total number of branches in the tree diagram.
For example, if the tree diagram has 9 branches in total and 6 of them have different colors for the two balls, then the probability of drawing two balls of different colors would be 6/9 or 2/3.