Puzzle - Red and Blue Marble

Puzzle

We have 50 Red , 50 Blue Marbles and 2 jars. One of the jar is chosen at random and then one marble is chosen from that jar at random. How would we maximize the chance of choosing red marble ? What is the probability of choosing red marble ?. All 100 marbles should be placed in the jar.

Solution

There are many combinations such that 100 marbles are in any one of the two jars. Now lets take a configuration 25 red and 25 blue marbles in one jar and the remaining 50 marbles in one jar.

P(choosing Red Marble)  = P(Jar 1) * P( Red Marble )  + P( Jar 2 ) * P( Red Marble)

= 0.5 * (25/50) + 0.5 *(25/50)

= 0.5

Let's try a different combination

We can have one red marble in one jar and the remaining 99 marbles in another jar.

P(choosing Red Marble)  = P(Jar 1) * P( Red Marble )  + P( Jar 2 ) * P( Red Marble)

= 0.5 * 1+ 0.5 *(49/99)

= 0.5 + 0.24747474747

=0.747474

I hope this is the correct answer. I don't have clear explanation how i arrived to one marble in the jar and remaining marbles in another jar. It was said to me by my friend Pandian