## Using Binomial Distribution Tables For p>0.5

The binomial tables are cumulative, so we only directly look up values of
$P(X \le x)$
for particular values of
$n, \: p$
. Normally binomial tabeles are given for values of
$p$
up and including
$p=0.5$
for various values of
$n$
. What do we do to find
$P(X \le 5)$
if
$X$
follows a
$B(10,0.6)$
distribution?
$p$
is usually taken to be the probability of a win or desirable outcome, and if this is greater than 0.5 then
$1-p$
, the probability of a loss or undesirable ouytcome is less than 0.5 and we can use the tables.
Remenbers thay
$Losses + WINS=n$
.
Find
$P(X \le 3)$
if
$X$
is modelled by
$B(10,0.6)$
.
The following rules apply.
$P(X \lt x) = P(X \le x-1)$

$P(X \gt x) = 1-P(X \le x)$

$P(X \ge x) = 1-P(X \le x-1)$

$WINS \le 3 \rightarrow LOSSES ge 7$

We find
$P(Y \ge 7$
using
$Y \sim B(10,0.4)$

$P(Y \ge 7)=1-P(Y \le 6)=1-0.9452=0.0548$