A previous Arizona state lottery called Lotto is played as...

A previous Arizona state lottery called Lotto is played as follows: The player selects six numbers from the numbers 1-42 and buys a ticket for $\$ 1 .$ There are six winning numbers, which are selected at random from the numbers 1-42. To win a prize, a Lotto ticket must contain three or more of the winning numbers. A probability distribution for the number of winning numbers for a single ticket is shown in the following table. $$\begin{array}{c|l} \hline \begin{array}{c} \text { Number of } \\ \text { winning numbers } \end{array} & \text { Probability } \\ \hline 0 & 0.3713060 \\ 1 & 0.4311941 \\ 2 & 0.1684352 \\ 3 & 0.0272219 \\ 4 & 0.0018014 \\ 5 & 0.0000412 \\ 6 & 0.0000002 \\ \hline \end{array}$$ a. If you buy one Lotto ticket, determine the probability that you win a prize. Round your answer to three decimal places. b. If you buy one Lotto ticket per week for a year, determine the probability that you win a prize at least once in the 52 tries.

A previous Arizona state lottery called Lotto is played as follows: The player selects six numbers from the numbers 1-42 and buys a ticket for $\$ 1 .$ There are six winning numbers, which are selected at random from the numbers 1-42. To win a prize, a Lotto ticket must contain three or more of the winning numbers. A probability distribution for the number of winning numbers for a single ticket is shown in the following table.
$$\begin{array}{c|l}
\hline \begin{array}{c}
\text { Number of } \\
\text { winning numbers }
\end{array} & \text { Probability } \\
\hline 0 & 0.3713060 \\
1 & 0.4311941 \\
2 & 0.1684352 \\
3 & 0.0272219 \\
4 & 0.0018014 \\
5 & 0.0000412 \\
6 & 0.0000002 \\
\hline
\end{array}$$
a. If you buy one Lotto ticket, determine the probability that you win a prize. Round your answer to three decimal places.
b. If you buy one Lotto ticket per week for a year, determine the probability that you win a prize at least once in the 52 tries.

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Key Concepts

Probability Distribution

A probability distribution is a function or table that provides the probabilities of occurrence of different possible outcomes in an experiment. It is central to the study of probability and statistics as it quantifies uncertainty and allows one to compute the likelihood of various events, such as winning different prizes in a lottery or obtaining other outcomes in random experiments.

Complement Rule in Probability

The complement rule is a fundamental concept in probability that states the probability of an event occurring is equal to one minus the probability of it not occurring. This is especially useful when calculating the probability of 'at least one' occurrence over multiple independent trials, since it can be simpler to determine the probability of no occurrences and subtract that from one.

Independent Events

Independent events are outcomes where the occurrence of one event does not influence the occurrence of another. This concept is critical when dealing with repeated experiments or trials, such as drawing lottery tickets over time, as it allows for the multiplication of individual probabilities to determine the overall likelihood of combined events.

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