[vsw id=”hu47ZbsskEw” source=”youtube” width=”611″ height=”344″ autoplay=”no”]<\/p>\n
Odds for and against an event represent a ratio of the desired outcomes versus the field. In other words, the odds for an event are the ratio of the number of ways the event can occur to the number of ways the event does not occur. Thus:<\/p>\n
<\/p>\n
Given the probability of an event \u2018E\u2019 i.e. P(E),<\/p>\n
$$ \\begin{align*}
\\text{Odds for E} & =\\cfrac {P(E)}{ \\left\\{ 1 \u2013 P(E) \\right\\} } \\\\
\\text{Odds against E} & = \\cfrac { \\left\\{ 1 \u2013 P(E) \\right\\} }{ P(E) } \\\\
\\end{align*} $$<\/p>\n
A box contains five blue balls, two green balls, and six yellow balls. What are the odds of drawing a blue ball from the box?<\/p>\n
First, we have to establish the probability of drawing a blue ball:<\/p>\n
Let P(B) represent the event that a blue ball is drawn from the box. Thus,<\/p>\n
$$ P(B) = \\cfrac {5}{13} $$<\/p>\n
$$ \\begin{align*}
\\text{The odds for a blue ball} & =\\cfrac {5}{13} \u00f7 \\left(1 \u2013 \\cfrac {5}{13} \\right)\\\\
& = \\cfrac {5}{13} \u00f7 \\left( \\cfrac {8}{13} \\right) \\\\
& =\\cfrac {5}{13} * \\cfrac {13}{8} \\\\
& =\\cfrac {5}{8} \\\\
\\end{align*} $$<\/p>\n
Therefore, the odds for a blue ball are 5:8 (pronounced as \u20185 to 8\u2019).<\/p>\n
Similarly, we can calculate the odds against drawing a blue ball:<\/p>\n
$$ \\begin{align*}
\\text{Odds against a blue ball} &= \\left\\{1 \u2013 \\cfrac{5}{13} \\right\\} \u00f7 \\cfrac{5}{13} \\\\
&= \\cfrac{8}{13} \u00f7 \\cfrac{5}{13} \\\\
&= \\cfrac{8}{13} * \\cfrac{13}{5} \\\\
& = \\cfrac{8}{5}
\\end{align*} $$<\/p>\n
Therefore, the odds against are 8:5 (pronounced as \u20188 to 5\u2019).<\/p>\n
You should notice that the odds against an event = reciprocal of odds for the same event.<\/em><\/p>\n Suppose you toss a fair coin. What are the odds against obtaining a head?<\/p>\n A. 2:2<\/p>\n B. 1:2<\/p>\n C. 1:1<\/p>\n Solution<\/strong><\/p>\n The correct answer is C.<\/p>\n The probability of obtaining a head, P(H) = 1\/2<\/p>\n Thus,<\/p>\n $$ \\begin{align*} Therefore, the odds against a head are 1:1, pronounced as \u20181 to 1.\u2019<\/p>\n Note that in this case, the odds for a head are also 1:1.<\/em><\/p>\n<\/blockquote>\n Reading 8 LOS 3c: describe the probability of an event in terms of odds for and against the event;<\/em><\/p>\n\n
Question<\/strong><\/h2>\n
\\text{Odds against a head} & = \\left\\{ 1 \u2013 P(H) \\right\\} \u00f7 P(H) \\\\
& = \\left\\{ 1 \u2013 \\cfrac{1}{2} \\right\\} \u00f7 \\cfrac{1}{2} \\\\
& = \\cfrac{1}{2} * \\cfrac{2}{1} \\\\
& = \\cfrac{1}{1}
\\end{align*} $$<\/p>\n