Step 1

Given that :

Binomial Probability \(\displaystyle=^{{{n}}}{C}_{{{x}}}\times{\left({p}\right)}^{{{x}}}\times{\left({q}\right)}^{{{n}-{x}}}\)

where \(\displaystyle{n}=\) number of trials and x is the number of successes.

Here \(\displaystyle{n}={24},{p}={0.2},{q}={1}-{p}={0.8}\)

Step 2

(a) P(at least 5) \(\displaystyle={P}{\left({5}\right)}+{P}{\left({6}\right)}+\ldots..+{P}{\left({25}\right)}\)

Using Technology P(At least 5) \(\displaystyle={0.5401}\)

(b) \(\displaystyle{P}{\left({5}\to{7}\right)}={P}{\left({5}\right)}+{P}{\left({6}\right)}+{P}{\left({7}\right)}={0.1960}+{0.1551}+{0.0997}={0.4508}\)

(c) \(\displaystyle{P}{\left(\text{At most}\ {6}\right)}={P}{\left({0}\right)}+{P}{\left({1}\right)}+\ldots.+{P}{\left({6}\right)}={0.8111}\)

Given that :

Binomial Probability \(\displaystyle=^{{{n}}}{C}_{{{x}}}\times{\left({p}\right)}^{{{x}}}\times{\left({q}\right)}^{{{n}-{x}}}\)

where \(\displaystyle{n}=\) number of trials and x is the number of successes.

Here \(\displaystyle{n}={24},{p}={0.2},{q}={1}-{p}={0.8}\)

Step 2

(a) P(at least 5) \(\displaystyle={P}{\left({5}\right)}+{P}{\left({6}\right)}+\ldots..+{P}{\left({25}\right)}\)

Using Technology P(At least 5) \(\displaystyle={0.5401}\)

(b) \(\displaystyle{P}{\left({5}\to{7}\right)}={P}{\left({5}\right)}+{P}{\left({6}\right)}+{P}{\left({7}\right)}={0.1960}+{0.1551}+{0.0997}={0.4508}\)

(c) \(\displaystyle{P}{\left(\text{At most}\ {6}\right)}={P}{\left({0}\right)}+{P}{\left({1}\right)}+\ldots.+{P}{\left({6}\right)}={0.8111}\)