A man alternately tosses a coin and throws a die beginning

Question

A man alternately tosses a coin and throws a die beginning with the coin. Find the probability that he gets a head in the coin and 5 or 6 in the die.

Bunuel · Answer

What does the word "alternately" has to do here? Does it matter he throws the coin first or not?

Please check the question and the source.

IF the question is about the probability of getting a head AND 5 or 6 in the die, then as these are independent events the probability would be 1/2*2/6=1/6. But again I doubt that the question testing the concept of independent event would be worded like this. So I suppose either this is very poor quality source or there is something else asked in the question. Maybe something is missing in the stem?

pradhan · Answer

Bunuel

I checked it from the source again. Its correct. The man tosses coin and die. While doing this he may get combination like H-5 or 5-H we need to consider the first one and discard the second. This is what I feel I am not sure. Nor the solution or OA is available. This question was asked in a University question paper.

Bunuel · Answer

With the current phrasing example provided doesn't make sense. If it were stated that  heads must appear before 5 or 6 , then yes, it would change everything and the answer in that case would be 3/4.

Meaning if it were: "A man alternately tosses a coin and throws a die beginning with the coin. Find the probability that he gets a head in the coin  before   5 or 6 in the die."

Answer: 3/4

Can you please provide with the link to this question?

pradhan · Answer

This question was asked in University of Roorkee, India exam. I will try to get the question paper or link. Till then lets hold on

roshanaslam · Answer

total sample size h1 h2 h3 h4 h5 h6 and t1 t2 t3 t4 t5 t6 ( head =H tail=T)
head with 5 and 6= 2 imes
prob2/12=1/6

by the way univ of roorkee its IIT rorkee

pradhan · Answer

roshanaslam

The probability of 1\6th is right if there is only one throw
if it is in the second throw then it will be (1\6)*(1\6)
in the third throw then it will be (1\6)*(1\6)*(1/6)

I am confused.

roshanaslam · Answer

A man  alternately tosses a coin and throws a die  beginning with the coin. Find the probability  that he gets a head in the coin and 5 or 6 in the die .

read the question again. he first tosses coin then the die. he never tossed the coin twice. i guess u got the total sample set. now sample set for H with 5 or 6=2(H5 and H6).

Bunuel · Answer

Let's consider the following problems:

1. Man tosses the fair coin and throws the die. What is the probability he gets a head in the coin AND 5 or 6 in the die?

The original question is similar to this one. The results of these two actions (tossing the coin and throwing the die) are independent events. The winning scenarios are {H,5} and {H,6}. Note that {5,H} is basically the same as {H,5}, the coin landed on heads and die gave 5.

The probability would be:  P(AandB)=P(A)*P(B)=
\frac{1}{2}*\frac{2}{6}=\frac{1}{6}

Answer:  \frac{1}{6}

NOTE:  Probability of A and B.
When two events are independent, the probability of both occurring is the product of the probabilities of the individual events: P(A and B) = P(A) x P(B)

2. Man tosses the fair coin and throws the die. What is the probability he gets a head in the coin OR 5 or 6 in the die?

Note this is  not  what the original question is asking.

In this case the probability would be:  P(AorB)=P(A)+P(B)-P(AandB)=\frac{1}{2}+\frac{2}{6}-\frac{1}{2}*\frac{2}{6}=\frac{2}{3}

Answer:  \frac{2}{3}

NOTE:  Probability of A OR B.
If Events A and B are independent, the probability that either Event A or Event B occurs is:
P(A or B) = P(A) + P(B) - P(A and B)

When we say "A or B occurs" we include three possibilities:
A occurs and B does not occur;
B occurs and A does not occur;
Both A and B occur.

3. Man tosses the fair coin and throws the die for n times. What is the probability he gets a head in the coin AND 5 or 6 in the die for all n times?

P=(\frac{1}{6})^n

So, if it were:  Man tosses the fair coin and throws the die for 3 times. What is the probability he gets a head in the coin AND 5 or 6 in the die 3 times?

Then simply:  \frac{1}{6}*\frac{1}{6}*\frac{1}{6}=\frac{1}{216}

Answer:  \frac{1}{216}

4. Man tosses the fair coin and throws the die for n times. What is the probability he gets a head in the coin AND 5 or 6 in the die k times?  (Meaning that the success is when during  one  try (toss&throw) he gets heads and 5 or 6. So, k success tries out of n.)

Probability of success is  \frac{1}{6}  as we calculated in the example 1.

NOTE:  If the probability of a certain event is p (1/6 in our case), then the probability of it occurring k times in n-time sequence is:

P = C^n_k*p^k*(1-p)^{n-k}

So if it were:  Man tosses the fair coin and throws the die for 3 times. What is the probability he gets a head in the coin AND 5 or 6 in the die 2 times? (Meaning that the success is when during  one  try (toss&throw) he gets heads and 5 or 6. So, two success tries out of 3.)

Then:  P =C^n_k*p^k*(1-p)^{n-k}= C^3_2*(\frac{1}{6})^2*(1-\frac{1}{6})^{(3-2)}=C^3_2*(\frac{1}{6})^2*\frac{5}{6}=\frac{5}{72}

Answer:  \frac{5}{72}

5. Man tosses the fair coin and throws the die for n times. What is the probability he gets a head m times in the coin AND 5 or 6 in the die k times?

P(H)=\frac{1}{2}  and  Q(5or6)=\frac{1}{3}

The same as in the previous example:  P = C^n_m*p^m*(1-p)^{n-m}*C^n_k*q^k*(1-q)^{n-k}=C^n_m*(\frac{1}{2})^m*(1-\frac{1}{2})^{n-m}*C^n_k*(\frac{1}{3})^k*(1-\frac{1}{3})^{n-k}= 
 =C^n_m*(\frac{1}{2})^m*(\frac{1}{2})^{n-m}*C^n_k*(\frac{1}{3})^k*(\frac{2}{3})^{n-k}

So if it were:  Man tosses the fair coin and throws the die for 5 times. What is the probability he gets a head 2 times in the coin AND 5 or 6 in the die 4 times?

P = C^5_2*(\frac{1}{2})^2*(\frac{1}{2})^{(5-2)}*C^5_4*(\frac{1}{3})^4*(\frac{2}{3})^{(5-4)}=C^5_2*(\frac{1}{2})^2*(\frac{1}{2})^{3}*C^5_4*(\frac{1}{3})^4*(\frac{2}{3})=\frac{25}{1944}

6. A man alternately tosses a coin and throws a die beginning with the coin. Find the probability that he gets a head in the coin BEFORE 5 or 6 in the die.

The one case when {H,5} and {5,H} is relevant is the above example. And more I think about this problem more I think this is what was intended to ask in the question. Here, I understand why the word "alternately" is used and I understand why the stem is stating that coin is tossed first.

Basically we have the following case man is tossing the coin and throws the die for several times. We are asked to calculate the probability that he gets "heads" BEFORE he gets 5 or 6.

Though I haven't seen GMAT question of this type, but here is solution:

The wining scenarios would be: 
{H, any value on die} meaning that during the first try he tossed and got the heads right away; 
OR
{T, any value but 5 or 6},{H, any value on die} meaning he didn't got heads and 5 or 6 during the first try but got heads during the second.
...

The probability would be the sum of the probabilities of such scenarios:

{H,Any}  +  {T, not 5 or 6}{H, Any}  +  {T, not 5 or 6}{T, not 5 or 6}{H, Any}  +  {T, not 5 or 6}{T, not 5 or 6}{T, not 5 or 6}{H, Any}  +  {T, not 5 or 6}{T, not 5 or 6}{T, not 5 or 6}{T, not 5 or 6}{H, Any}  +....=

=\{\frac{1}{2}*1\}+\{\frac{1}{2}*\frac{2}{3}\}*\{\frac{1}{2}*1\}+\{\frac{1}{2}*\frac{2}{3}\}\{\frac{1}{2}*\frac{2}{3}\}*\{\frac{1}{2}*1\}+\{\frac{1}{2}*\frac{2}{3}\}\{\frac{1}{2}*\frac{2}{3}\}\{\frac{1}{2}*\frac{2}{3}\}*\{\frac{1}{2}*1\}...+...=\frac{1}{2}+\frac{1}{2}*\frac{1}{3}+\frac{1}{2}*\frac{1}{3}*\frac{1}{3}+\frac{1}{2}*\frac{1}{3}*\frac{1}{3}*\frac{1}{3}+...

As you can see we have the geometric progression with first term  b_1=\frac{1}{2}  and common ratio  r=\frac{1}{3} . When 0<common ration<1, then the sum of geometric progression equals to  \frac{b_1}{1-r} .

Hence the above sum equals to  \frac{\frac{1}{2}}{1-\frac{1}{3}}=\frac{\frac{1}{2}}{\frac{2}{3}}=\frac{3}{4}

Answer:  \frac{3}{4}

Hope it's clear.

pradhan · Answer

Bunuel

Thanks a lot .... the concept is now clear

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A man alternately tosses a coin and throws a die beginning

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