Respuesta :
Answer:
1/16
Step-by-step explanation:
The probabilty of choosing a red is 3 out of 12 or
1/4
. Since you replace it, there are still 3 red balls and 12 total balls left, so that probability is also
1/4
. Multiply
1/4
and
1/4
and the correct answer is
1/16
.
Probability of an event is the measure of chance of occurrence of that event. The probability of the considered event is 0.0625
What is the chain rule in probability for two events?
For two events A and B:
The chain rule states that the probability that A and B both occur is given by:
[tex]P(A \cap B) = P(A)P(B|A) = P(B)P( A|B)[/tex]
If two events A and B are independent, then we get
P(A|B) = P(A) and P(B|A) = P(B), thus getting [tex]P(A \cap B) = P(A)P(B)[/tex]
How to calculate the probability of an event?
Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.
Then, suppose we want to find the probability of an event E.
Then, its probability is given as
[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}}[/tex]
Where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.
For the given case, let we consider two events as:
- A = event of picking a red ball in first draw
- B = event of picking a red ball in second draw
Since after first draw, there is replacement of the drawn ball, thus, keeping the collection of ball complete. That makes A and B independent of each other(theoretically, if collection is considered to be fully randomly shuffled in both the draws).
The probabilities of both the events are same as the collection is unchanged and they are about drawing a red ball out of the given collection ,
Probability of drawing a red ball = (number of ways of drawing red)/(total number of ways of drawing a ball)
P(A) = 3/12 (since 3 red balls, so [tex]^3C_1 = 3[/tex] ways of selecting red ball
and [tex]^{12}C_1=12[/tex] ways of selecting a ball out of all balls.
The needed probability is probability of event A and B both to occur.
This is denoted symbolically as [tex]P(A \cap B)[/tex]
Using the chain rule, we get:
[tex]P(A \cap B) = P(A)P(B) \text{\:\:(Since both events are independent)}\\\\P(A \cap B) = \dfrac{3}{12} \times \dfrac{3}{12} = 0.0625[/tex]
Thus, the probability of the considered event (back to back getting red ball if selection is with replacement) is 0.0625
Learn more about probability here:
brainly.com/question/1210781
