Question

At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many are expected to not to use the foil as their main weapon?

e. Find the probability that six do not use the foil as their main weapon.

f. Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically.

Short answer

a. The random variable X is the number of fencers who do not use the foil as their main weapon.

b. The values that X may take on are $0,1,2,......,25$

c. The distribution $X~B(25,0.40)$

d. $10$are not expected to use the foil as their main weapon.

e. The probability that six do not use the foil as their main weapon is $0.0442$

f. It would be very surprising as the probability of all $25$that did not use the foil is zero.

Step-by-step solution

Content Introduction

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

Part (a) Step 1: Explanation

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.

Part (b) Step 1: Explanation

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:

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

Part (c) Step 1: Explanation

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,

$X~B(25,0.40)$

Part (d) Step 1: Explanation

The expected binomial distribution is calculated as:

$\mu =np$, where,

$\mu$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,

$$\begin{gathered} \mu =np \\ \mu =25\times 0.40 \\ \mu =10 \end{gathered}$$

Part (e) Step 1: Explanation

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

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

Part (f) Step 1: Explanation

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

Therefore, it is quite surprising.