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A common daily lottery game involves the drawing of three digits from 0 to 9 independently with replacement and independently from day to day. Lottery watchers often get excited when all three digits are the same, an event called triples.

If p is the probability of obtaining triples, and if X is the number of days without triples before the first triple is observed, then X has the geometric distribution with parameter p. In this case, it is easy to see that p = 0.01, since there are 10 different triples among the 1000 equally likely daily numbers.

The number X of daily draws without a tripleuntil we see a triple has the geometric distribution with parameter p = 0.01. \(X \sim Geo\left( {p = 0.01} \right)\)

a.

The probability that two particular days in a row will both have triples is the same as having two consecutive successes.

\(\begin{array}{c}p\left( x \right) = {\left( {0.1} \right)^2}\\ = 0.0001\end{array}\)

Thus, the required probability is 0.0001.

b.

The conditional probability is defined as follows:

\(P\left( {A|B} \right) = \frac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}\)

It is read as Probability A, given B is probability of A and B upon probability of occurrence of B.

In this case,

A=occurrence of a triple the next day

B=occurrence of a triple today

Therefore,

\(\begin{array}{c}P\left( {triple\;tomorrow|triple\;today} \right) = \frac{{P\left( {triple\;today\;and\;tomorrow} \right)}}{{P\left( {triple\;today} \right)}}\\ = \frac{{0.01 \times 0.01}}{{0.01}}\\ = 0.01\end{array}\)

Thus, the conditional probability that we observe triples again the next day given they occurred today is 0.01.