Question

English and American spellings are colour and color, respectively. A man staying at a Parisian hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40 percent of the English-speaking men at the hotel are English and 60 percent are Americans, what is the probability that the writer is an Englishman?

Answer 1

Answer:

The probability is 5/11

Step-by-step explanation:

Let's call V the event that the letter taken at random is a vowel.

Let's call E the event that the man is English and A to the event that the man is American.

If 40 percent of the English-speaking men at the hotel are English means P(E)=0.40 and 60 percent are Americans means P(A)=0.60

In ''color'' we have 2 vowels out of 5 letters so P(V/A)=2/5

In ''colour'' we have 3 vowels out of 6 letters so P(V/E)=3/6=1/2

P(E/V)=P(E∩ V)/P(V)

P(V)=P(V|E)P(E) +P(V|A)P(A)

P(V)=(1/2)0.40+(2/5)0.60=0.44

P(E∩V)=P(V|E)P(E)=(1/2)0.40=0.20

P(E/V)=0.20/0.44=0.45454545=5/11

Answer 2

Probability of an event is the measure of its occurrence. The probability that the writer is an Englishman is 0.45 approx.

How to calculate the probability of an event?

Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.

Then, suppose we want to find the probability of an event E.

Then, its probability is given as

$P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}}$

where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.

What is chain rule in probability?

For two events A and B, by chain rule, we have:

$P(A \cap B) = P(B)P(A|B) = P(A)P(A|B)$

For the given case, we can take the things in terms of event. Taking the given things as events, we get:

• A = Event of that person writing the word being English
• B = Event of that person writing that word being an American

• C = Event of picking a vowel from the word that person writes.

Then, as it is given that 40 percent of the English-speaking men at the hotel are English and 60 percent are Americans,

thus, P(A) = 40% = 0.40

and P(B) = 60% = 0.60

Also, using the conditional probability, we get that:

P(C|A) = 3/6 = 0.5 (C|A means A is given, it means it is specified that the word written is "colour" It has 3 vowels and 6 total letters)

similarly, we get P(C | B) = 2/5 = 0.4

Now, using the chain rule, we get:

$P(A \cap C) = P(A) \times P(C|A) = 0.4 \times 0.5 = 0.2\\P(B \cap C) = P(B) \times P(B|A) = 0.6 \times 0.4 = 0.24$

The needed probability is that probability that the writer is an Englishman

given that the letter taken at random was found to be vowel.

It is written symbolically as: $P(A|C)$

Using the law of total probability, we got:

$P(C) = P(A \cap C) + P(B \cap C) = 0.2 + 0.24 = 0.44$

Thus, using the chain rule, we get:

$P(A|C)\times P(C) = P(A \cap C)\\ P(A | C) = \dfrac{P(A \cap C) }{P(C)} = \dfrac{0.2}{0.44} \approx 0.45$

Thus, the probability that the writer is an Englishman is 0.45 approx.

Learn more about probability here:

brainly.com/question/1210781