Actually I'm asking this question because I found my answer for the HW question somewhat weird, but I still can't determine where did I make mistakes.
Given: A basketball player can score on a particular shot with p = 0.3. Let S be the number of successes out of 25 independent shots.
Question: Find approximate probability that S <= 5, then compare to the exact probability
First of all, I think this is a case of Binomial distribution, am I right?
** Here is my work:
P(S <=5)= P(Z <= [5 - n(mean)] / [(standard deviation)* (n^(1/2))]
where I use:
mean = np = 25 * 0.3 = 7.5
sd = [np(1-p)]^(1/2) = 2.29
After doing all the jobs on calculator, I get P(Z <= -1.09) = 1 - P(Z >= 1.09) = 0.1379
** However, when I do P(S = 5), using the probability mass function of Binomial distribution
P(S = 5) = (25C5)(.3^5)(.7^15) = 0.612
which is way off from my estimation P(S <= 5). My bet is that the two numbers have to be somewhat close.
Please help me to check if I did make errors or miss something. Now I'm really confused on trying to figure this out.
Thank you
I believe your first P(Z) number using the normal distribution.
You need to compare that with the sum of the exact probabilities of getting 0, 1, 2, 3, 4 and 5. using the binomial distribution.
P(0) = (0.7)^25 = 0.000134
P(1) = (0.7)^24*(0.3) *25!/(24!*1!)
= 0.001437
P(2) = (0.7)^23*(0.3)^2* [24*25/(1*2)]
= 0.007390
P(5) = (0.7)^20*(0.3)^5*[25!/(20!*5!)]
= 0.10302
You do the others, P(3) and P(4), and add them all. You might be quite close to the normal distribution result.
It looks to me like you did P(5) incorrectly
Based on your description, it appears that you are on the right track with using the binomial distribution for this problem. However, there seems to be some misunderstanding in your calculation.
To find the approximate probability that S <= 5, you can use the normal approximation to the binomial distribution. However, there are a few corrections to be made.
1. Calculating the mean:
You correctly calculated the mean as np which is 25 * 0.3 = 7.5.
2. Calculating the standard deviation:
The formula you used for the standard deviation is correct. The formula is √(np(1-p)). However, you made an error in calculating it. The correct calculation should be √(25 * 0.3 * 0.7) = 2.24.
3. Calculating the Z-score:
To find the Z-score, you need to subtract the mean from the value (5) and divide it by the standard deviation. So the correct calculation is (5 - 7.5) / 2.24 ≈ -1.12.
4. Calculating the approximate probability:
To find the approximate probability that S <= 5, you can use the standard normal distribution table or a calculator to find the probability associated with the Z-score -1.12. From the standard normal distribution table, the probability is approximately 0.1314.
Comparing this to the exact probability that S = 5, which you calculated correctly as 0.612, we can see that they are indeed not close. This suggests that the normal approximation may not be very accurate for this specific case.
In conclusion, it seems that you made a mistake in calculating the standard deviation, which led to an incorrect Z-score and approximate probability.