(11.4) Two balls are to be selected with replacement from a bag that contains one red, one blue, one green, and one orange ball. Determine the probability that the orange ball is selected at least once.
This answer makes no sense.
)selected at least once) = 1 - (not selected) = 1 - (.75 * .75)
To determine the probability that the orange ball is selected at least once when two balls are selected with replacement, we need to consider two scenarios:
Scenario 1: The orange ball is selected on the first draw, and any ball is selected on the second draw.
Scenario 2: The orange ball is not selected on the first draw, but it is selected on the second draw.
Step 1: Analyze Scenario 1
In scenario 1, the probability of selecting the orange ball on the first draw is 1/4, as there is one orange ball out of four total balls.
Since we are selecting with replacement, the probability of selecting any ball on the second draw is also 1/4. The orange ball is already back in the bag, so we don't have to consider it again.
To calculate the probability of both events occurring together, we multiply the individual probabilities:
P(Scenario 1) = P(Orange on 1st draw) * P(Any on 2nd draw) = (1/4) * (1/4) = 1/16
Step 2: Analyze Scenario 2
In scenario 2, the probability of not selecting the orange ball on the first draw is 3/4 since there are three non-orange balls out of four total balls.
Again, because we are selecting with replacement, the probability of selecting the orange ball on the second draw is 1/4, as the orange ball is back in the bag.
To calculate the probability of both events occurring together, we multiply the individual probabilities:
P(Scenario 2) = P(Not Orange on 1st draw) * P(Orange on 2nd draw) = (3/4) * (1/4) = 3/16
Step 3: Calculate the Probability of the Orange Ball Being Selected at Least Once
To find the probability that the orange ball is selected at least once, we need to consider both scenarios. We can add their probabilities together:
P(Orange ball selected at least once) = P(Scenario 1) + P(Scenario 2) = 1/16 + 3/16 = 4/16 = 1/4
Therefore, the probability that the orange ball is selected at least once when two balls are selected with replacement is 1/4.
To determine the probability that the orange ball is selected at least once when two balls are selected with replacement from the bag, we need to find the probability of two scenarios:
1. The orange ball is selected once
2. The orange ball is selected twice
Let's calculate the probability of each scenario separately and then add them together.
1. Probability of selecting the orange ball once:
- There is only one orange ball in the bag.
- The total number of balls in the bag is four.
- After taking out one orange ball, there are still four balls left to choose from.
- Therefore, the probability of selecting the orange ball once is 1/4.
2. Probability of selecting the orange ball twice:
- The orange ball is put back into the bag after each selection, so it is available for selection again in the second draw.
- The probability of selecting the orange ball in each draw is 1/4.
- Since the selections are independent events, we multiply the probabilities together: (1/4) * (1/4) = 1/16.
Now, we can add the probabilities of the two scenarios to get the overall probability of selecting the orange ball at least once:
1/4 + 1/16 = 5/16
Therefore, the probability of selecting the orange ball at least once when two balls are selected with replacement is 5/16.