There is a large container which holds 5 red balls, 3 green balls and 2 purple balls. The balls are all the same size and texture. You are required to select two balls.
If both of the balls that you select are purple then you win.
How much less likely are you to win if you are required to replace the first ball? Show all of your work and explain your steps.
If we are required to replace the first ball, it means that after selecting a ball, we put it back into the container before selecting the second ball. This means that for each ball selection, we have the same chance of selecting any ball, regardless of the previous selection.
In this case, there are a total of 5 red balls, 3 green balls, and 2 purple balls.
The probability of selecting a purple ball on the first selection is 2 purple balls out of a total of 10 balls:
P(purple on the first selection) = 2/10 = 1/5
Since we replace the first ball, the probability of selecting a purple ball on the second selection is also 2 purple balls out of a total of 10 balls:
P(purple on the second selection) = 2/10 = 1/5
To calculate the probability of both selections being purple, we multiply the probabilities:
P(both purple with replacement) = P(purple on the first selection) * P(purple on the second selection)
= (1/5) * (1/5)
= 1/25
Therefore, the probability of winning by selecting 2 purple balls with replacement is 1/25.
To find out how much less likely you are to win if you are required to replace the first ball, we can compare it to the probability of winning without replacement.
When we don't replace the first ball, the total number of balls decreases by one for the second selection. After removing the first purple ball, there will be 1 purple ball remaining out of 9 balls.
The probability of selecting a purple ball on the second selection without replacement is 1 purple ball out of a total of 9 balls:
P(purple on the second selection without replacement) = 1/9
To calculate the probability of both selections being purple without replacement:
P(both purple without replacement) = P(purple on the first selection) * P(purple on the second selection without replacement)
= (1/5) * (1/9)
= 1/45
Therefore, the probability of winning by selecting 2 purple balls without replacement is 1/45.
To determine how much less likely you are to win if you are required to replace the first ball, we take the ratio of these two probabilities:
(1/45) / (1/25) = (1/45) * (25/1) = 25/45 = 5/9
Therefore, you are 5/9 times less likely to win if you are required to replace the first ball.