Step 1

Given Information:

Suppose that you flip a coin 14 times.

To find the probability that you achieve at least 4 tails:

Let X be a random variable which denotes the number of tails and X follows Binomial distribution with number of trials \(\displaystyle{n}={14}\) and probability of success \(\displaystyle{p}={0.5}\)

(Since, \(\displaystyle{P}{\left({T}\right)}={\frac{{{1}}}{{{2}}}}={0.5}\))

Probability mass function of Binomial variable X is given by the formula:

\(\displaystyle{P}{\left({X}={x}\right)}=^{{{n}}}{C}_{{{x}}}{p}^{{{x}}}{\left({1}-{p}\right)}^{{{n}-{x}}};{x}={0},{1},{2}\ldots.{.14}\)

Step 2

Required probability is obtained as follows:

\(\displaystyle{P}{\left({X}\geq{4}\right)}={1}-{P}{\left({X}{ < }{4}\right)}\)

\(\displaystyle={1}-{\left[{P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}+{P}{\left({X}={2}\right)}+{P}{\left({X}={3}\right)}\right]}\)

\(\displaystyle={1}-{\left[^{\left\lbrace{14}\right\rbrace}{C}_{{{0}}}{\left({0.5}\right)}^{{{0}}}{\left({1}-{0.5}\right)}^{{{14}-{0}}}+^{{{14}}}{C}_{{{1}}}{\left({0.5}\right)}^{{{1}}}{\left({1}-{0.5}\right)}^{{{14}-{1}}}+^{{{14}}}{C}_{{{2}}}{\left({0.5}\right)}^{{{2}}}{\left({1}-{0.5}\right)}^{{{14}-{2}}}+^{{{14}}}{C}_{{{3}}}{\left({0.5}\right)}^{{{3}}}{\left({1}-{0.5}\right)}^{{{14}-{3}}}\right]}\)

\(\displaystyle={1}-{\left[{\frac{{{14}!}}{{{0}!{\left({14}-{0}\right)}!}}}\times{1}\times{0.00006103515625}+{\frac{{{14}!}}{{{1}!{\left({14}-{1}\right)}!}}}\times{0.5}\times{0.0001220703125}+{\frac{{{14}!}}{{{2}!{\left({14}-{2}\right)}!}}}\times{0.25}\times{0.000244140625}+{\frac{{{14}!}}{{{3}!{\left({14}-{3}\right)}!}}}\times{0.125}\times{0.00048828125}\right]}\)

\(\displaystyle={1}-{\left[{1}\times{0.00006103515625}+{14}\times{0.00006103515625}+{91}\times{0.00006103515625}+{364}\times{0.00006103515625}\right]}\)

\(\displaystyle={1}-{\left[{0.00006103515625}+{0.0008544921875}+{0.005554199219}+{0.02221679688}\right]}\)

\(\displaystyle={1}-{0.02868652344}\)

\(\displaystyle={0.9713134766}\)

\(\displaystyle\approx{0.9713}\)

Thus, the probability that you achieve at least 4 tails is 0.9713

Given Information:

Suppose that you flip a coin 14 times.

To find the probability that you achieve at least 4 tails:

Let X be a random variable which denotes the number of tails and X follows Binomial distribution with number of trials \(\displaystyle{n}={14}\) and probability of success \(\displaystyle{p}={0.5}\)

(Since, \(\displaystyle{P}{\left({T}\right)}={\frac{{{1}}}{{{2}}}}={0.5}\))

Probability mass function of Binomial variable X is given by the formula:

\(\displaystyle{P}{\left({X}={x}\right)}=^{{{n}}}{C}_{{{x}}}{p}^{{{x}}}{\left({1}-{p}\right)}^{{{n}-{x}}};{x}={0},{1},{2}\ldots.{.14}\)

Step 2

Required probability is obtained as follows:

\(\displaystyle{P}{\left({X}\geq{4}\right)}={1}-{P}{\left({X}{ < }{4}\right)}\)

\(\displaystyle={1}-{\left[{P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}+{P}{\left({X}={2}\right)}+{P}{\left({X}={3}\right)}\right]}\)

\(\displaystyle={1}-{\left[^{\left\lbrace{14}\right\rbrace}{C}_{{{0}}}{\left({0.5}\right)}^{{{0}}}{\left({1}-{0.5}\right)}^{{{14}-{0}}}+^{{{14}}}{C}_{{{1}}}{\left({0.5}\right)}^{{{1}}}{\left({1}-{0.5}\right)}^{{{14}-{1}}}+^{{{14}}}{C}_{{{2}}}{\left({0.5}\right)}^{{{2}}}{\left({1}-{0.5}\right)}^{{{14}-{2}}}+^{{{14}}}{C}_{{{3}}}{\left({0.5}\right)}^{{{3}}}{\left({1}-{0.5}\right)}^{{{14}-{3}}}\right]}\)

\(\displaystyle={1}-{\left[{\frac{{{14}!}}{{{0}!{\left({14}-{0}\right)}!}}}\times{1}\times{0.00006103515625}+{\frac{{{14}!}}{{{1}!{\left({14}-{1}\right)}!}}}\times{0.5}\times{0.0001220703125}+{\frac{{{14}!}}{{{2}!{\left({14}-{2}\right)}!}}}\times{0.25}\times{0.000244140625}+{\frac{{{14}!}}{{{3}!{\left({14}-{3}\right)}!}}}\times{0.125}\times{0.00048828125}\right]}\)

\(\displaystyle={1}-{\left[{1}\times{0.00006103515625}+{14}\times{0.00006103515625}+{91}\times{0.00006103515625}+{364}\times{0.00006103515625}\right]}\)

\(\displaystyle={1}-{\left[{0.00006103515625}+{0.0008544921875}+{0.005554199219}+{0.02221679688}\right]}\)

\(\displaystyle={1}-{0.02868652344}\)

\(\displaystyle={0.9713134766}\)

\(\displaystyle\approx{0.9713}\)

Thus, the probability that you achieve at least 4 tails is 0.9713