Question

question 28
if the odds against a certain event are 3 to 4, what is the probability of the event occuring? write your
answer as a simplified fraction.
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question 29
if the probability of a certain event is \\( \frac { 71 } { 74 } \\), what are the odds against the event happening?
write all answers as simplified fractions.
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Explanation:

Question 28

Step1: Recall the formula for probability from odds against

If the odds against an event \(E\) are \(a:b\), then \(P(E)=\frac{b}{a + b}\). Here \(a = 3\) and \(b=4\).

Step2: Calculate the probability

Substitute \(a = 3\) and \(b = 4\) into the formula \(P(E)=\frac{b}{a + b}\). So \(P(E)=\frac{4}{3+4}\).

Question 29

Step1: Find the probability of the event not - happening

If \(P(E)=\frac{71}{74}\), then \(P(\text{not }E)=1 - P(E)\). Using the formula \(P(\text{not }E)=1-\frac{71}{74}=\frac{74 - 71}{74}=\frac{3}{74}\).

Step2: Calculate the odds against the event

The odds against an event \(E\) is given by \(\frac{P(\text{not }E)}{P(E)}\). Substitute \(P(E)=\frac{71}{74}\) and \(P(\text{not }E)=\frac{3}{74}\) into the formula. \(\frac{P(\text{not }E)}{P(E)}=\frac{\frac{3}{74}}{\frac{71}{74}}\). Since \(\frac{\frac{a}{c}}{\frac{b}{c}}=\frac{a}{b}\) (for \(c
eq0\)), the odds against is \(\frac{3}{71}\), or written as \(3:71\).

Answer:

Question 28: \(\frac{4}{7}\)
Question 29: \(3:71\)