homework help. please and thank you!! 2. Cell phones use different color codes w

Question

homework help. please and thank you!!

2. Cell phones use different color codes when crossing from the coverage area of one base station to another. During a study, two different models of cell phones were used on a driving trip to record how many times each color code was used as they switched coverage areas. A table is shown below of all the times the phones accessed each color code L05 b cl Total Model 1 20 35 40 95 Model 2 15 50 6 475 Total 35 85 46 170 Suppose we randomly select one of the two cell phones at a point during the trip. a) What is the probability that the cell phone is Model I, given that it is using color code 5? b) What is the probability that the cell phone is using either color code 0 or b, given that it is model 17 c) What is the probability that the cell phone is not using color code c given that it is model 17 d) Are the events "using color code O and "the phone is Model 2" independent? 3. In a class of 260 seniors, 93 study Spanish, 95 study Chemistry, 165 study Mathematics, 18 study Spanish and Chemistry, 75 study Chemistry and Math, 20 study Math and Spanish and 15 study all three subjects. Suppose we choose one of the seniors at random. Use this information to answer the following questions. a) What is the probability that the chosen student doesn't study any of these subjects? b) What is the probability that the chosen student doesn't study Spanish, given that they study mathematics? c) What is the probability that the chosen student studies chemistry, given that they study mathematics or Spanish? d) What is the probability that the chosen student studies both mathematics and chemistry, given that they don't study Spanish?

Explanation / Answer

Answer to question# 2)

Part a)

We need to find the probability that the cell phone is of model I , we laready know that it is using color code 5

Thus this implies we need to find the conditional probablity of moel I given code 5

.

Formula of conditional probability

P(A |B) = P(A given B) = P(A and B) / P(B)

.

LIkewise we get:

P(Model I | code 5) = Probability of mdel I given code 5

P(model I | code 5) = P(model I AND Code 5) / P(Code 5)

.

Formula of probability = favorable outcomes / total outcomes

.

P(code 5) = total outcomes with code 5 / total outcomes

P(Code 5) = (35+50) / 170

P(code 5) = 85 /170

.

P(model I and Code 5) = total outcomes with model I and code 5 / total number of outcomes

P(Model I and Code 5) = 35 /170

.

On plugging the values of P(Code 5) and P(model I and Code 5) in the formula of conditional probability we get

P(Model I | Code 5) = (35 /170) / (85 /170)

P(Model I | Code 5) = 35 /85

P(Model I | Code 5) = 0.4118

.

Part b)

We need to find the probability that the phone uses either code 0 or b , and it is already given that the phone is of model I

Again , this is question is asking about conditional probability

P(Code 0 or b | Model I) = Probability of code 0 or B given it is model I

.

Thus using the conditional formula we get

P(Code 0 or b | Model I) = P((code 0 or b) AND Model I) / P(Model I)

.

Now P((Code 0 or b) and Model I) = P(code 0 and model I or Code b and model I)

P(code 0 and model I or code b and model I) = P(code 0 and model I) + P(code b and model I)

.

P(code 0 and model I) = number of outcomes with code 0 and model I / total number of outcomes

P(code 0 and model I) = 20 /170

.

P(code b and model I) = number of outcomes with code b and model I / total number of outcomes

P(code b and model I) = 40 /170

.

Thus P(code 0 or b AND model I) = (20 /170) + (40/170)

P(code 0 or b AND model I) = 60 /170

.

P(model I) = number of outcomes with model I / total number of outcomes

P(model I) = (20 +35 +40 +0) / 170

P(model I) = 95 /170

.

On plugging these values in the formula of conditional probablity we get:

P(Code 0 or b | Model I) = (60 /170) / (95/170)

P(code 0 or b | model I ) = 60 /95 = 0.6316

.

Part c)

We need to find the probability that the phone is not using code c given that it is model I

Again this question has the condition that the phone is model I

P(No Code c | Model I) = 1 - P(Code c |model I)

[rule of complementary events]

.

Using the formula of conditional probability we get

P(code c | model I) = P(code c AND model I) / P(model I)

.

P(code c and model I) = number of outcomes with code c and model I / total number of outcomes

P(code c and model I) = 0 /170

.

P(model I) = number of outcomes with model I / total number of outcomes

P(model I) = 95 /170

.

On plugging the vlaues in formula of conditional probability we get

P(code c | model I) = 0

.

P(No code c | model I) = 1 - 0 = 1

Thus porbability that the model has no code c , given that it is model I is 1

.

Part d)

If two events are independent then the multiplication rule must apply on them

As per the multiplication rule

P(A and B) = P(A) *P(B)

then the events A and B are considered to be independent events

.

Let us verify this equation for the two events : Code 0 and model II

P(Code 0 and Model II) = Number of outcomes with code 0 and model II / total number of outcomes

P(code 0 and model II) = 15 /170

.

P(code 0) = number of outcomes with code ) / total number of outcomes

P(code 0) = (20 +15) / 170

P(code 0) = 35 /170

.

P(model II) = number of outcomes with model II / total number of outcomes

P(model II) = (15 +50 +6 +4) / 170

P(model II) = 75 /170

.

NOw we got LHS = P(model II and code 0) = 15/170

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Now we check the right hand side of the multiplication rule , and check if it equates to LHS or not

RHS = P(Model II) * P(code 0)

RHS = (75 /170) * (35 /170)

RHS = (15/34) * (7 /34)

RHS = 105 / 1156 is not equal LHS

.

Hence the multipication rule for Model II and code 0 does not hold true

Hence the events Model II and Code 0 are NOT independent.

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