Question

In a school, three subject teachers English, Math and Science sometimes give surprise tests on the same day. Based on past records:

• The English teacher gives a test 90% of the time.
• The Math teacher gives a test 80% of the time.
• The Science teacher gives a test 70% of the time.

Each teacher decides independently. If the average number of surprise tests is less than 2.3 then the teachers should coordinate better to increase the performance of the students. Otherwise, no action is needed.

Let X be the number of surprise tests a student gets on a given day. So, X ∈ {0, 1, 2, 3}.

1. Find the probability for each possible number of surprise tests. 
2. Use the probabilities to build a distribution table. 
3. Calculate the average number of surprise tests per day. 
4. Based on your calculations, decide: Should the teachers coordinate better? Or is the current plan acceptable?

Answer 1

i. Let `P(E) = 9/10, P(M) = 4/5` and `P(S) = 7/10`

⇒ `P(barE) = 1/10, P(barM) = 1/5` and `P(barS) = 3/10`

`P(X = 0) = 1/10 xx 1/5 xx 3/10`

`P(X = 0) = 3/500`

`P(X = 1) = (9/10 xx 1/5 xx 3/10) + (1/10 xx 4/5 xx 3/10) + (1/10 xx 1/5 xx 7/10)`

`P(X = 1) = 46/500`

`P(X = 2) = (9/10 xx 4/5 xx 3/10) + (9/10 xx 1/5 xx 7/10) + (1/10 xx 4/5 xx 7/10)`

`P(X = 2) = 199/500`

`P(X = 3) = 9/10 xx 4/5 xx 7/10`

`P(X = 3) = 252/500`

ii. 

X	0	1	2	3
P(X)	`3/500`	`46/500`	`199/500`	`252/500`

iii. Average number of surprise tests = E(X)

= `0 + (1 xx 46/500) + (2 xx 199/500) + (3 xx 252/500)`

= `1200/500`

= 2.4

iv. Average number of surprise tests = 2.4 > 2.3

Students are getting a good average of surprise tests.

So, the current system is balanced and the teachers do not need to change anything.