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The binomial distribution determines the probability of looking at a specific quantity of a hit results in a specific quantity of trials.

We are given,

$60\%$of the fencers use the foil as their main weapon and $25$fencers are at The Fencing Center.

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case the random variable X is the number of fencers who do not use the foil as their main weapon.

Make the list of values that you want to use X may take on.

As we can see there is an upper bound for the situation at hand $25$then X is given by:

$X=0,1,2,......,25.$

The random variable is distributed by the data provided X being the number of fencers who do not use the foil as their main weapon.

According to the given information $60\%$of the fencers use the foil as their main weapon and $25$fencers are at The Fencing Center.

So, here the percentage of the fencers who do not use the foil as their main weapon is$40\%$

The probability distribution of binomial distribution has two parameters role="math" localid="1649083966344" $$\begin{gathered} n=25 \\ isnumberoftrials \\ p=0.40 \\ isprobabilityofsuccess \end{gathered}$$

The binomial distribution is of the form: $X~B(n,p)$

Therefore,

$X~B(25,0.40)$

The expected binomial distribution is calculated as:

$渭=np$, where,

$渭$is the number of fencers who are expected to not to use the foil as their main weapon,

role="math" localid="1649084286946" $n=25$is number of trials

role="math" localid="1649084294410" $p=0.40$is probability of the fencers who do not use the foil as their main weapon .

Therefore,

$$\begin{gathered} 渭=np \\ 渭=25脳0.40 \\ 渭=10 \end{gathered}$$

The probability that $6$ do not use the foil as their main weapon is as follow:

$=\left(\begin{array}{c}25!\\ 6!19!\end{array}\right){(0.4)}^6{(0.6)}^{19}=0.442$

The probability that all $25$do not use the foil as main weapon is Zero.

Therefore, it is quite surprising.