Solution

Let $R,Y$ and  $B$ be the event that red, yellow and blue colour, respectively, is chosen by a subject.

$$B

$$R, ~Y

Let $abc$ be the event that $a, ~b$ and $c$ are chosen by subject in the experiment.

Let $x$ be the number of times red colour is chosen.

a. Since, all the colours were flashed for the same time, subject would be randomly choosing one of the three colours.

$$P(R)=P(Y)=P(B)=\frac{1}{3}$$

$$$$\begin{align*} P(x=0)&= P(R^c R^c R^c )\\ &= P(R^c)^3\\ &=\left(\dfrac{2}{3}\right)^3\\\\ P(x=1)&= P(R R^c R^c )+ P(R^c R R^c )+ P(R^c R R^c )\\ &= 3 \left(\dfrac{1}{3} \left( \dfrac{2}{3} \right)^2\right)\\ &= \left(\dfrac{2}{3}\right)^2\\\\ P(x=2 )&= P(R R R^c )+ P(R R^c R )+ P(R^c R R )\\ &= 3 \left(\dfrac{2}{3} \left( \dfrac{1}{3} \right)^2\right)\\ &=2 \left(\dfrac{1}{3}\right)^2\\\\ P(x=3)&= P(RRR)\\ &=P(R)^3\\ &= \left(\dfrac{1}{3}\right)^3 \end{align*}

$$P(R)= P(Y)= P(B) = \dfrac{1}{3}$$

To calculate the probabilities, we use multiplication rule and the fact that three choices are independent.

b. Following is the probability histogram for random variable, $x$:

Answer

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