Question

Of 240 students, 176 are on the honor roll, 48 are members of the varsity team, and 36 are in the honor roll and are also members of the varsity team. What is the probability that a randomly selected student is on the honor roll or is a member of the varsity team?

WRONG ANSWER = REPORTED​

Answer 1

`\blue{\huge {\mathrm{PROBABILITY}}}`

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`{\underline{\huge \mathbb{Q} {\large \mathrm {UESTION : }}}}`

• Of 240 students, 176 are on the honor roll, 48 are members of the varsity team, and 36 are in the honor roll and are also members of the varsity team. What is the probability that a randomly selected student is on the honor roll or is a member of the varsity team?

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`{\underline{\huge \mathbb{A} {\large \mathrm {NSWER : }}}}`

• The probability that a randomly selected student is on the honor roll or is a member of the varsity team is
  `\boxed{\bold{\:(47)/(60)\:}}`

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`{\underline{\huge \mathbb{S} {\large \mathrm {OLUTION : }}}}`

We can use the inclusion-exclusion principle to find the number of students who are on the honor roll or are members of the varsity team.

This principle states that:

• 
  `\sf |A\cup B| = |A| + |B| − |A\cap B|`

where:

• A and B are sets,
• |A| is the cardinality (number of elements) of set A, and
• A∩B is the intersection of sets A and B.

Using this principle, we can find that:

`\begin{aligned}\sf |Honors\cup Varsity|& =\sf |Honors| + |Varsity| − |Honors\cap Varsity|\\& =\sf 176 + 48 - 36\\& =\sf\red{188}\end{aligned}`

Therefore, there are 188 students who are on the honor roll or are members of the varsity team.

The probability that a randomly selected student is on the honor roll or is a member of the varsity team is then:

`\begin{aligned}\sf P(Honors\cup Varsity)& =\sf (|Honors\cup Varsity|)/(|Total|) \\ &=\sf (188)/(240) \\&=\boxed{\bold{\: (47)/(60)\:}}\end{aligned}`

Therefore, the probability that a randomly selected student is on the honor roll or is a member of the varsity team is
`\boxed{\bold{\:(47)/(60)\:}}`

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`- \large\sf\copyright \: \large\tt{AriesLaveau}\large\qquad\qquad\qquad\qquad\qquad\qquad\tt 04/02/2023`

Answer 2

47/60

Explanation:

You want to know the probability of a randomly selected student is on the honor roll or varsity team when 176 of 240 students are on the honor roll, 48 are on the varsity team, and 36 are on both.

One or the other

The probability of A or B is ...

P(A+B) = P(A) +P(B) - P(A·B)

The probability of interest is ...

P(honor roll + varsity) = P(honor roll) + P(varsity) - P(honor roll & varsity)

P(honor roll + varsity) = 176/240 +48/240 -36/240 = (176 +48 -36)/240

= 188/240 = 47/60

The probability of interest is 47/60.