In the Small State Lottery,
three white balls are drawn at random from twenty balls labled 1-20 and
a blue Superball is drawn from ten balls labeled 21-30.
To win a prize you must match at least two of the white balls or match the blue Superball.
If you buy a ticket what is the probability that you would win a prize?
Let W be the probability of choosing 2 white balls out of 20 or 3 white balls out of 20 .
Let \(\overline{\mathbf{W}}\)be not W.
Let B be the probability of choosing one blue ball out of 10.
Let \(\overline{\mathbf{B}}\) be not B
\(\begin{array}{|rcll|} \hline W &=& \dfrac{\binom{3}{2}\binom{17}{1} + \binom{3}{3}\binom{17}{0} }{\binom{20}{3}} = \dfrac{13}{285} \\\\ \overline{W} &=& 1 - W = 1-\dfrac{13}{285} = \dfrac{272}{285} \\\\ B &=& \dfrac{\binom{1}{1}\binom{9}{0} }{\binom{10}{1}} = \dfrac{1}{10} \\\\ \overline{B} &=& 1 - B = 1 - \dfrac{1}{10} = \dfrac{9}{10} \\ \hline \end{array} \)
\(\begin{array}{|c|c|c|c|} \hline & \mathbf{B} & \mathbf{\overline{B}} \\ & \dfrac{1}{10} & \dfrac{9}{10} \\ \hline \mathbf{W} & W\cap B & W\cap \overline{B} \\ \dfrac{13}{285} & \dfrac{13}{285} \times \dfrac{1}{10} & \dfrac{13}{285} \times \dfrac{9}{10} \\ \hline \mathbf{\overline{W}} & \overline{W}\cap B & \overline{W}\cap \overline{B} \\ \dfrac{272}{285} & \dfrac{272}{285} \times \dfrac{1}{10} & \dfrac{272}{285} \times \dfrac{9}{10} \\ \hline \end{array} \)
The probability of not a prize is
\(\overline{W}\cap \overline{B} = \dfrac{272}{285} \times \dfrac{9}{10} \\\)
The probability of a prize is
\(\begin{array}{|rcll|} \hline && 1- \overline{W}\cap \overline{B} \\ &=& 1- \left(\frac{272}{285} \times \frac{9}{10} \right) \\ &=& \frac{285\cdot 10- 9\cdot 272}{285\cdot 10} \\ &=& \frac{402}{2850} \\ &=& \frac{67\cdot 6}{475\cdot 6} \\ &=& \frac{67}{475} \\ &=& 0.14105263158 \\ \hline \end{array} \)
