A box contains 4 red and 3 blue poker chips. What is the probability when 3 are selected randomly that all 3 wil be red if we select chips with replacement?

a.) Without replacement?

To get one red out of the seven, you have 4 chances, so the probability is

4/7.
Since the chips will be replaced, the second and subsequent draws will be under the same conditions, i.e. they are independent. The probability of each subsequent draw is again 4/7.
The probability of joint independent events is the product of the individual probabilities, thus
in this case,
(4/7)*(4/7)*(4/7)=?

Without replacement, after each successful draw, there will be one red chip less.
Probability for first draw:
4/7 as before
second draw:
3/6
third draw:
2/5
So probability of all three events happening:
(4/7)*(3/6)*(2/5)
=4/35

A box contains 7 White and 4 Red poker chips.

What is the probability when 3 are selected randomly without replacement that they will all be white?

Well, selecting chips with replacement means that each chip you select is put back into the box before the next selection. So, in this scenario, when selecting chips with replacement, the probability of selecting a red chip on each draw stays the same, since the total number of chips in the box does not change.

To calculate the probability of selecting all 3 red chips with replacement, we can simply multiply the individual probabilities of selecting a red chip on each draw.

The probability of selecting a red chip on the first draw is 4/7 (since there are 4 red chips out of a total of 7 chips).

The probability of selecting a red chip on the second draw is also 4/7, since we put the previously selected chip back into the box.

The probability of selecting a red chip on the third draw is also 4/7.

So, the probability of selecting all 3 red chips with replacement is (4/7) * (4/7) * (4/7) = 64/343.

Therefore, the probability is 64/343.

To find the probability of selecting 3 red chips with replacement, we need to determine the probability of selecting a red chip on each individual draw.

The probability of selecting a red chip in one draw is given by the number of red chips divided by the total number of chips in the box. In this case, there are 4 red chips and 3 blue chips, so the probability of selecting a red chip on any given draw is 4/7.

Since each draw is done with replacement, the probability of selecting a red chip on each of the three draws is (4/7) * (4/7) * (4/7) = 64/343. Therefore, the probability of selecting all three red chips when three are selected randomly with replacement is 64/343.

Now let's consider the case without replacement. In this case, after each draw, the chip is not replaced back into the box, so the total number of chips decreases for each subsequent draw.

For the first draw, the probability of selecting a red chip is 4/7, as before. However, for the second draw, there are now only 6 chips left in the box, with 3 being red and 3 being blue. Therefore, the probability of selecting a red chip on the second draw is 3/6, which simplifies to 1/2.

Similarly, for the third draw, there are now only 5 chips left in the box, with 2 being red and 3 being blue. Therefore, the probability of selecting a red chip on the third draw is 2/5.

To find the probability of all three draws being red, we multiply the individual probabilities together: (4/7) * (1/2) * (2/5) = 8/70, which simplifies to 4/35.

Therefore, the probability of selecting all three red chips when three are selected randomly without replacement is 4/35.

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