Probability and Statistics Lecture 13

Continued

Probability and Statistics Lecture 12

Prerequisites

• Binomial Distribution

Purely solved questions this class

Questions

Question 1

There are $5$ Defective items and $20$ good Items. Find the probability Distribution of the defective items if 4 items are drawn from the lot.

Answer 1

Number of items = $25$

Question 0

Two Unbiased Die rolled 3 times find prob that sum 9 is obtained once and twice

Question 3

Suppose theres 2000 computer chips in a batch and there is a 2% prob that any one chip is faulty. Find the probability that 5 of them are faulty

Question 4

Precision Bombing attack there is a 50% chance that any 1 bomb will strike the target. two direct hits are required to destroy the target completely how many bomb must be dropped at least 99% chance of destroying the bomb.w

Question 5

The probability that an archer strikes a target is 1/3 If he fires five times whats the probability of hitting 2 times. This is a binomial probability problem, where we use the binomial distribution formula:

$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

where:

• $n=5$ (number of trials)
• $k=2$ (number of successes)
• $p=\frac{1}{3}$ (probability of hitting the target)
• $1-p=\frac{2}{3}$ (probability of missing)

Step 1: Compute the Binomial Coefficient

$$\left(\genfrac{}{}{0ex}{}{5}{2}\right)=\frac{5!}{2!(5-2)!}=\frac{5!}{2!3!}=\frac{5\times 4}{2\times 1}=10.$$

Step 2: Compute the Probability Terms

$p^k={\left(\frac{1}{3}\right)}^2=\frac{1}{9}$

$(1-p{)}^{n-k}={\left(\frac{2}{3}\right)}^3=\frac{8}{27}$

Step 3: Compute the Final Probability

$$P(X=2)=10\times \frac{1}{9}\times \frac{8}{27}$$ $$=10\times \frac{8}{243}$$ $$=\frac{80}{243}$$

Approximating:

$\frac{80}{243}\approx 0.3296$

Final Answer:

$$P(X=2)\approx 0.33\text{ (or }\frac{80}{243}\text{ exactly)}$$

References

Continued

Probability and Statistics Lecture 14

Information

• date: 2025.02.11
• time: 15:11