Question

Two teaching methods, A and B, are implemented for learning Spanish. There is a 70% chance of successfully learning Spanish if method A is used, and a 85% chance of success if methodB is used. However, method B is substantially more 3. time consuming and is therefore used only 20% of the time (method A is used the other 80% of the time). The following notations are suggested:

A-Method A is used.
B-Method B is used.
L-Spanish was learned successfully Interpret each of the probabilities given in the original problem in probability
a) notation with their respective values (four of these).
b) Draw a Probability Tree representing each of the probabilities mentioned above.
c) What is the likelihood a person does not learn Spanish?
d) A student learned the spanish successfully. What is the probability that they were taught by method B?

Answer 1

(a) Shown below.

(b) Shown below.

(c) The probability that a person does not learn Spanish is 0.18.

(d) The probability that method B was used given that a student learned the Spanish successfully is 0.83.

Explanation:

(a)

Consider the provided data.

`P(A)=0.20\\P(B)=0.80\\P(L|A)=0.70\\P(L|B)=0.85`

Then the probability of not learning Spanish using the respective methods are:

`P(L'|A)=1-P(L|A)=1-0.70=0.30\\\\P(L'|B)=1-P(L|B)=1-0.70=0.15`

(b)

The Probability Tree representing each of the probabilities mentioned above is attached below.

(c)

Compute the probability that a person does not learn Spanish as follows:

`P(L')=P(L'|A)P(A)+P(L'|B)P(B)`

`=(0.30* 0.20)+(0.15* 0.80)\\\\=0.06+0.12\\\\=0.18`

Thus, the probability that a person does not learn Spanish is 0.18.

(d)

Compute the probability that method B was used given that a student learned the Spanish successfully as follows:

`P(B|L)=(P(L|B)P(B))/(P(L|A)P(A)+P(L|B)P(B))`

`=((0.85* 0.80))/((0.70* 0.20)+(0.85* 0.80))\\\\=(0.68)/(0.14+0.68)\\\\=0.82927\\\\\approx 0.83`

Thus, the probability that method B was used given that a student learned the Spanish successfully is 0.83.