Probability is the chance of occurrence of an event. The probability that both marbles are drawn by the Cymra are red is [tex]\dfrac{25}{196}[/tex].
The probability helps us to know the chances of an event occurring.
[tex]\rm{Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]
Given to us
The bag contains 5 red marbles,6 green marbles, and 3 blue marbles.
We need to find the probability of both the marbles picked by Cymra to be red, in order to do that we will find the probability of Cymra picking a red marble for one time.
[tex]\rm Probability(Red) = \dfrac{\text{Number of red marbles in the bag}}{\text{Number of marbles in the bag}}[/tex]
Substitute the values,
[tex]\rm Probability(Red) = \dfrac{5}{14}[/tex]
Now as it is given that Cymra put the red marble back in the bag, therefore, the sample size will be the same and the number of red marble in the bag will also be equal, therefore, the probability of picking a red marble will be,
[tex]\rm Probability(Red) = \dfrac{\text{Number of red marbles in the bag}}{\text{Number of marbles in the bag}}[/tex]
Substitute the values,
[tex]\rm Probability(Red) = \dfrac{5}{14}[/tex]
Now, the probability that both marbles are drawn by the Cymra are red,
[tex]\rm Probability(2\ Red) = \dfrac{5}{14} \times \dfrac{5}{14} = \dfrac{25}{196}[/tex]
Hence, the probability that both marbles are drawn by the Cymra are red is [tex]\dfrac{25}{196}[/tex].
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