Suppose that in a certain school the probability that a student studies Chinese is 0.22, the probability that a student studies French is 0.35, and the probability that a student studies both languages is 0.14. Find the probability that a student in that school studies:
a) at least one of these languages
b) exactly one of these languages.
a) To find the probability that a student studies at least one of these languages, we need to find the probability of the union of the two events: studying Chinese and studying French. We can use the formula:
P(Chinese or French) = P(Chinese) + P(French) - P(Chinese and French)
Substituting the given probabilities, we get:
P(Chinese or French) = 0.22 + 0.35 - 0.14 = 0.43
Therefore, the probability that a student in that school studies at least one of these languages is 0.43.
b) To find the probability that a student studies exactly one of these languages, we need to subtract the probability of studying both languages from the probability of studying at least one language. That is:
P(exactly one language) = P(Chinese or French) - P(Chinese and French)
Substituting the given probabilities, we get:
P(exactly one language) = 0.43 - 0.14 = 0.29
Therefore, the probability that a student in that school studies exactly one of these languages is 0.29.
To find the probability that a student in the school studies at least one of the languages, we need to calculate the probability of the union of the two events, i.e., when a student studies Chinese or French or both.
a) Probability of studying at least one language:
P(Chinese or French) = P(Chinese) + P(French) - P(Chinese and French)
P(Chinese or French) = 0.22 + 0.35 - 0.14
= 0.57
Therefore, the probability that a student in the school studies at least one of these languages is 0.57.
b) To find the probability that a student in the school studies exactly one language, we need to consider the probabilities of the following events:
1. Studying Chinese only (not studying French)
2. Studying French only (not studying Chinese)
b) Probability of studying exactly one language:
P(Chinese only) = P(Chinese) - P(Chinese and French)
= 0.22 - 0.14
= 0.08
P(French only) = P(French) - P(Chinese and French)
= 0.35 - 0.14
= 0.21
The probability that a student in the school studies exactly one of these languages is the sum of the probabilities of studying Chinese only and French only:
P(Chinese or French only) = P(Chinese only) + P(French only)
= 0.08 + 0.21
= 0.29
Therefore, the probability that a student in the school studies exactly one of these languages is 0.29.
To find the probabilities, we can use the principles of probability and set theory.
a) To find the probability that a student in the school studies at least one of these languages, we need to calculate the probability of the union of the events: P(Chinese or French).
To do this, we can use the formula:
P(Chinese or French) = P(Chinese) + P(French) - P(Chinese and French)
Given information:
P(Chinese) = 0.22
P(French) = 0.35
P(Chinese and French) = 0.14
Substituting the values into the formula:
P(Chinese or French) = 0.22 + 0.35 - 0.14
Calculating:
P(Chinese or French) = 0.43
Therefore, the probability that a student in that school studies at least one of these languages is 0.43.
b) To find the probability that a student in the school studies exactly one of these languages, we need to calculate the probability of the exclusive events: P(Chinese but not French) + P(French but not Chinese).
To do this, we can use the formula:
P(Chinese but not French) = P(Chinese) - P(Chinese and French)
P(French but not Chinese) = P(French) - P(Chinese and French)
Given information:
P(Chinese) = 0.22
P(French) = 0.35
P(Chinese and French) = 0.14
Substituting the values into the formula:
P(Chinese but not French) = 0.22 - 0.14
P(French but not Chinese) = 0.35 - 0.14
Calculating:
P(Chinese but not French) = 0.08
P(French but not Chinese) = 0.21
Now, to find the probability of exactly one of these languages:
P(exactly one language) = P(Chinese but not French) + P(French but not Chinese)
Substituting the calculated values:
P(exactly one language) = 0.08 + 0.21
Calculating:
P(exactly one language) = 0.29
Therefore, the probability that a student in that school studies exactly one of these languages is 0.29.