## 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?We think about losses instead of wins!

\[p\]

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 \[Y \sim B(10,0.4)\]

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