You select a marble without looking and then put it back. If you do this 6 times, what is the best prediction possible for the number of times you will pick a purple or a pink marble?

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Let $$ p $$ be the probability of picking a purple or pink marble in one trial. Since you replace the marble after each pick, each draw is independent. The number of times you pick a purple or pink marble in 6 trials follows a binomial distribution.

1. Define the number of trials:  
   $$
   n = 6
   $$

2. The expected value (or best prediction) for a binomial random variable is given by:  
   $$
   E(X) = n \times p
   $$

3. If we assume that the marbles are equally likely to be any color, and that purple and pink together make up half of the possibilities, then:  
   $$
   p = \frac{1}{2} = 0.5
   $$

4. Therefore, the expected number of times you will pick a purple or pink marble is:  
   $$
   E(X) = 6 \times 0.5 = 3
   $$

So, the best prediction is $$ 3 $$.
The best prediction is that you will pick a purple or pink marble 3 times out of 6 tries.

You select a marble without looking and then put it back. If you do this 6 times, what is the best prediction possible for the number of times you will pick a purple or a pink marble?

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