An electrical system has the switches shown in the following diagram to indicate

Question

An electrical system has the switches shown in the following diagram to indicate system operation based on the switch position. The probability that Switch A is closed is: P[A_closed] = 7/9 The probability the Switch B selecting the Red LED or the Order LEDs is: P[B_Red] = 2/5 P[B_Other] = 3/5 The probability that the Selector switch S selects the various colored LEDs is: P[S_Blue] =4/13 P[S_Yellow] = 2/13 P[S_Green] = 3/13 P[S_White] = 4/13 a. What is the probability that all the LEDs are off? b. What is the probability that the White LED is on? c. What is the probability that either the Red LED or the Green LED is on? d. Given that the Switch B is set to the "B_Other" LEDs position, what is the probability that either the Blue LED or the Yellow LED is on?

Explanation / Answer

a) all LEDs are off when A is opened irrespective of positions of B and switch S

probability that all LEDs are off = P[A_opened] = 1- P[A_closed] = 1-(7/9) = 2/9

b) white LED is on when switch A is closed and Switch B is connected to other switch s is connected to white

p[white_LED_on] = p[A_closed]*p[B_other]*p[S_white] = ( 7/9)*(3/5)*(4/13) = 84/585

c) probability that either Red LED or green LED is on

p[RUG] = p[Red_on]+p[Green_on] -p[R and G]

red and green cannot be on at the same time =>mutually exclusive events so, p[R and G] = 0

p[RUG] = p[Red_on]+p[Green_on]

p[Red_on] = p[A_closed]*p[B_red] = 7/9*2/5 = 14/45 red is on when A is closd and B is connected red

p[Green_on] = p[A_closed]*p[B_other]*p[s_green] = 7/9*3/5*3/13 = 63/585 green is on when A is closed and B is

connected to other and s is connected to green

p[R U G] = 14/45 + 63/585

d) B is set to B_other

p[Blue U Yellow] = p[Blue] + p[Yellow] - p[Blue and Yellow]

p[Blue and Yellow] = 0 here Blue Yellow cannot glow at the same time

Blue glows when A is closed and S is connected to Blue, B is already connected to B_other

p[Blue] = p[A_closed]*p[S_blue] = 7/9*4/13 = 28/117

p[Yellow] = p[A_closed]*p[S_yellow] = 7/9*2/13 = 14/117

p[Blue U Yellow] = p[Blue] + p[Yellow] = 28/117 + 14/117 = 42/117

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