I have got 18 red , 12 green, 5 blue and 5 yellow balls in the bag.
What is the probability that I pick excatly 2 green when I pick 4 at random ?
What is the probabily that none are yellow after I pick 4 at random ?
Thanks.
total, N = 40
Assume no replacement,
P(G1)=12/40
P(G2)=11/39
P(G1∩G2)
=12*11/(40*39)
=11/130
35 balls are not yellow (X)
So if all are not yellow, follow the same principle as above,
P(X1)*P(X2)*P(X3)*P(X4)
=(35/40)*(34/39)*(33/38)*(32/37)
=...
Use hypergeometric distribution as Reiny showed you in your previous problem:
C(12,2)*C(28,2)/C(40,4)=12474/45695
where we need to choose 2 out of 12 greens, 2 out of 28 non-greens, and in total, 4 out of 40 balls.
For the second part, you can use what was given above, OR use the hypergeometric distribution again:
P(4 non-yellow)=C(5,0)*C(35,4)/C(40,4)
again, we are choosing 0 out of 5 yellows, 4 out of 35 non-yellows, and 4 out of 40 in all.
Both methods give 5236/9139.
Note that for probability calculations, fractions give exact answers, while decimals give approximate and sometimes ambiguous answers.
To find the probability of picking a specific number of green balls, we need to know the total number of possible outcomes (the denominator) and the number of favorable outcomes (the numerator). Let's calculate both probabilities step-by-step:
1. Probability of picking exactly 2 green balls when picking 4 at random:
First, determine the total number of possible outcomes. Since you have a total of 18 + 12 + 5 + 5 = 40 balls in the bag, the total number of ways to pick 4 balls is given by the combination formula: 40 choose 4, denoted as C(40, 4), which is calculated as:
C(40, 4) = 40! / (4! * (40 - 4)!) = 91,390
Next, calculate the number of favorable outcomes. We need to choose exactly 2 green balls from the 12 available, which can be done in C(12, 2) ways:
C(12, 2) = 12! / (2! * (12 - 2)!) = 66
Therefore, the probability of picking exactly 2 green balls when picking 4 at random is:
P(2 green) = favorable outcomes / total outcomes = C(12, 2) / C(40, 4) = 66 / 91,390 ≈ 0.00072
So, the probability is approximately 0.00072 or 0.072%.
2. Probability of none of the 4 balls being yellow:
Again, start by finding the total number of possible outcomes. Since there are 40 balls in the bag, the number of ways to choose 4 balls is given by C(40, 4), which is still 91,390.
Next, calculate the number of favorable outcomes, which is the number of ways to choose 4 balls from the available non-yellow balls (18 red + 12 green + 5 blue) in the bag:
C(18 + 12 + 5, 4) = C(35, 4) = 35! / (4! * (35 - 4)!) = 52,360
Therefore, the probability of none of the 4 balls being yellow is:
P(no yellow) = favorable outcomes / total outcomes = C(35, 4) / C(40, 4) = 52,360 / 91,390 ≈ 0.573
So, the probability is approximately 0.573 or 57.3%.