Question

An automobile service facility specializing in engine tune-ups knows that 25% of tune-ups are done for four-cylinder engines, 40% for six-cylinder engines, and 35% for eight- cylinder engines. The facility does 100 tune-ups per day unless it runs out of sparkplugs. Let X be number of sparkplugs the facility needs in a day
1. What is the approximate) PDF of X?
2. If the facility has 600 sparkplugs in stock at the beginning of the day and there are no new sparkplug shipments received during the day, what is the probability that it will run out by the end of the day?

Answer 1

Answer:

1.  620

2. 2.5%

Step-by-step explanation:

X = No of spark plugs needed by the facility in a day

Now, for tuning a 4 cylinder engine, we need 4 spark plugs for one engine. In total, we need 400 spark plugs for 100 tune ups

For tuning a 6 cylinder engine, we need 6 spark plugs for one engine. In total, we need 600 spark plugs for 100 tune ups

for tuning a 8 cylinder engine, we need 8 spark plugs for one engine. In total, we need 800 spark plugs for 100 tune ups

Engine           X P(X)  X*P(X) Cf

4 Cylinder 400 25%    100 100

6 Cylinder 600 40%   240  340

8 Cylinder 800 35%   280  620

1. Therefore , the approximate total demand of spark plugs = 620

2. Now, we have 600 spark plugs, which will cater to 100 units(25%) of 4 cylinders demand, 240 units (40%) of 6 cylinder demand, 260 units of 8 cylinder.

The prob that spark plugs will meet 8 cylinder demand = 35% or around an estimated 280 spark plugs

i.e. 280 spark plugs meets 35% of the demand.

So, 260 spark plugs will meet (35/28 * 26) % = 32.5% of the demand.

Therefore, total demand met =  25% +  40% + 32.5% = 97.5%

Therefore, prob that it will run out of spark plugs by the end of the day = 1 - 97.5% = 2.5%