a) The class has thirty students. Twenty students study physics while twelve study both physics and chemistry. If eight students study neither physics nor chemistry, how many stude... a) The class has thirty students. Twenty students study physics while twelve study both physics and chemistry. If eight students study neither physics nor chemistry, how many students study chemistry? b) The knock-out match is the one which emerges with a winner, no draw match. The football experts predict the outcome of two knock-out matches. The first match is of Simba against Yanga, the second is of Real Madrid against Barcelona. The chance of Simba to win is 0.74 while that of Real Madrid to win is 0.65. What is the probability that both Yanga and Barcelona will win their matches?
Understand the Problem
The image contains two probability questions. The first question requires finding the number of students studying chemistry given information about students studying physics, both, or neither subject. The second question involves calculating the probability of two independent events (Yanga winning and Barcelona winning) occurring together, given their individual win probabilities.
Answer
Number of students studying chemistry: $14$ Probability that both Yanga and Barcelona will win their matches: $0.091$
Answer for screen readers
Number of students studying chemistry: $14$
Probability that both Yanga and Barcelona will win their matches: $0.091$
Steps to Solve
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Calculate the number of students studying at least one subject
Subtract the number of students studying neither subject from the total number of students.
$30 - 8 = 22$
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Determine the number of students studying only physics
Subtract the number of students studying both subjects from the number of students studying physics.
$20 - 12 = 8$
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Calculate the number of students studying only chemistry
Subtract the number of students studying only physics and the number of students studying both subjects from the total number of students studying either physics, chemistry or both.
$22 - 8 - 12 = 2$
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Find the number of students studying chemistry
Add the number of students studying only chemistry to the number of students studying both chemistry and physics.
$2 + 12 = 14$
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Calculate the probability of Yanga winning
Since the probability of Simba winning is $0.74$, and only one of them can win, the probability of Yanga winning is $1 - 0.74$.
$P(\text{Yanga wins}) = 1 - 0.74 = 0.26$
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Calculate the probability of Barcelona winning
Since the probability of Real Madrid winning is $0.65$, the probability of Barcelona winning is $1 - 0.65$.
$P(\text{Barcelona wins}) = 1 - 0.65 = 0.35$
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Calculate the probability of both Yanga and Barcelona winning
Since the two events are independent, multiply their individual probabilities.
$P(\text{Yanga wins and Barcelona wins}) = P(\text{Yanga wins}) \times P(\text{Barcelona wins}) = 0.26 \times 0.35 = 0.091$
Number of students studying chemistry: $14$
Probability that both Yanga and Barcelona will win their matches: $0.091$
More Information
The first question is a pretty standard venn diagram question while the second requires the application of probability rules relating to independent events.
Tips
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