Probability: Calculating the Chances of Success

[Rojito]

(a) If the probability of hitting a target is ¾ (0.75), find the probability of hitting the target first on the fourth shot.

More generally, if the probability of hitting the target is P, what is the probability of hitting the target for the first time on the nth shot?

(b) A computer system in a large financial institution is based on 4 mainframe machines. Each machine has its own emergency power supply and operates independently. The machines are dedicated to the following tasks:

(i) Administration
(ii) Back-up Administration System
(iii) Customer Database
(iv) Back-up Customer Database

The probability of each machine failing is 0.03. The system operates properly if both the administrative and customer database machines can be supported. What is the probability that the system functions properly?

(c) The probability that a salesperson makes a sale to a customer on a first visit is 0.3. If no sale is made, the salesperson calls again the following week and the probability of making a sale then is 0.45. If no sale is made and the salesperson calls for a third and last time the probability of a sale is 0.2. Find the probability that the salesperson will make a sale to a particular customer.

 

[Rojito]

Would I be correct in assuming that the chance of a mishit is .25
\therefore 3 mishits = 1/4.1/4.1/4=1/64
The next attempt (4th shot) = 3/4.1/64=3/256

 

[soroban]

Hello, Rojito!

Would I be correct in assuming that the chance of a miss is 1/4 ?
Therefore, 3 misses = (1/4)(1/4)(1/4) = 1/64
The next attempt (4th shot) = (3/4)(1/64) = 3/256

Correct!

 

[Rojito]

Thanks Soroban,

I was hoping for some help on the rest of the question, which I'm totally stuck on.

 

[CellCoree]

(a) The probability of hitting the target first on the fourth shot can be calculated using the formula P(A) = (1-P)^3 * P, where P is the probability of hitting the target (0.75 in this case). This gives us a probability of approximately 0.0586, or 5.86%.

More generally, the probability of hitting the target for the first time on the nth shot can be calculated using the formula P(A) = (1-P)^(n-1) * P.

(b) The probability of a single machine failing is 0.03, so the probability of all four machines functioning properly is (1-0.03)^4 = 0.8835. However, the system only operates properly if both the administrative and customer database machines are functioning, so the overall probability is 0.8835 * 0.8835 = 0.7801, or approximately 78.01%.

(c) The probability of making a sale on the first visit is 0.3, and the probability of making a sale on the second visit is 0.45. If no sale is made on the first two visits, the probability of making a sale on the third visit is 0.2. Therefore, the overall probability of making a sale to a particular customer can be calculated by adding the probabilities of making a sale on each visit, but subtracting the probability of making a sale on all three visits (since this would mean the salesperson did not make a sale at all). This gives us a probability of 0.3 + 0.45 + 0.2 - (0.3 * 0.45 * 0.2) = 0.735, or approximately 73.5%.