An urn contains 5 red balls and 10 white balls. 6 balls are randomly drawn and not replaced. What is the probability that 3 or fewer balls drawn are red?
could be 3 red, 3 white
could be 2 red, 4 white
could be 1 red, 5 white
could be all white
prob of those taken in order from top to bottom
= C(6,3) (5/15)(4/14)(3/13)(10/12)(9/11)(8/10)
+ C(6,2) (5/15)(4/14)(10/13)(9/12)(8/11)(7/10)
+ C(6,1) (5/15)(10/14)(9/13)(8/12)(7/11)(6/10)
+ C(6,0) (10/15)(9/14)(8/13)(7/12)(6/11)(5/10)
= .... you do the button pushing
Can you see the pattern ?
The answer is supposed to be 0.04 + 0.25 + 0.42 + 0.24 = 0.95
However, when I do the math above, I get 0.01 + 0.03 + 0.04 + 0.04 = 0.12.
Where am I making the error in calculating this?
To determine the probability that 3 or fewer balls drawn are red, we need to consider all the possible outcomes and count the favorable outcomes (i.e., the outcomes where 3 or fewer balls are red) and divide it by the total number of possible outcomes.
There are a couple of ways to approach this problem. One way is to consider the cases where exactly 0, 1, 2, or 3 red balls are drawn.
Case 1: 0 red balls drawn
In this case, all 6 balls drawn must be white. The number of ways to choose 6 white balls from a set of 10 is given by the combination formula: C(10, 6).
Case 2: 1 red ball drawn
We need to choose 1 red ball and 5 white balls. The number of ways to choose 1 red ball from 5 and 5 white balls from 10 is given by: C(5, 1) * C(10, 5).
Case 3: 2 red balls drawn
We need to choose 2 red balls and 4 white balls. The number of ways to choose 2 red balls from 5 and 4 white balls from 10 is given by: C(5, 2) * C(10, 4).
Case 4: 3 red balls drawn
We need to choose 3 red balls and 3 white balls. The number of ways to choose 3 red balls from 5 and 3 white balls from 10 is given by: C(5, 3) * C(10, 3).
Now, we can calculate the probabilities for each case:
P(case 1) = C(10, 6) / (C(15, 6))
P(case 2) = (C(5, 1) * C(10, 5)) / (C(15, 6))
P(case 3) = (C(5, 2) * C(10, 4)) / (C(15, 6))
P(case 4) = (C(5, 3) * C(10, 3)) / (C(15, 6))
Finally, we add up these probabilities to get the final probability:
P(3 or fewer red balls drawn) = P(case 1) + P(case 2) + P(case 3) + P(case 4)
You can calculate these combinations and probabilities using a scientific calculator, or by using software like Python with libraries such as numpy and scipy.