Help The Policeman in Finding The Culprit

Late one evening, a car ran over a pedestrian in a narrow bystreet and drove away without stopping. A policeman who saw the vehicle leave the scene of the accident reported it moving at very high speed. The accident itself was witnessed by six bystanders. They provided the following conflicting accounts of what had happened:

• It was a blue car, driven by a man;
• The car was moving at high speed, its headlights were turned off;
• The car did have license plates, it wasn’t going very fast;
• It was a Toyota, it’s headlights were turned off;
• The car didn’t have license plates, the driver was a woman;
• It was a gray Ford.

When the the car and its driver were finally apprehended, it turned out that only one of the six eyewitnesses gave a fully correct description. Each of the other five provided one true and one false piece of information.

Keeping that in mind, can you determine the following:

— What was the car’s brand?

— What color was the car?

— Was the car going fast or slow?

— Did it have license plates?

— Were its headlights turned on?

— Was the driver a man or a woman? 

Read all the answers here!

To Help The Policeman in Finding The Culprit

But why he needs help? 

Let's recollect all the statements made by all 6 bystanders.

 1.It was a blue car, driven by a man.

 2.The car was moving at high speed, its headlights were turned off.

 3.The car did have license plates, it wasn’t going very fast.

 4.It was a Toyota, it’s headlights were turned off.

 5.The car didn’t have license plates, the driver was a woman.

 6 It was a gray Ford (It was gray car; it was Ford).

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If we believe in report made by Policeman where he stated that the car was moving at very high speed; then the part of the Statement 3 made by third bystander where he says car wasn't going fast turns out to be false. Hence, other part of his statement must be true. So the car must have license plates.

If the car has license plates; then 1st part of the Statement 5 will be false & other part must be true. Hence, the driver must be a woman.

If the driver was a woman, then 2nd part of the Statement 1 turns false making part 1 to be true. Hence, the color of the car must be blue.

If the car was at high speed then the entire Statement 2 must be true or it's 2nd part must be false.

Let's assume 2nd part of the statement 2 be false. Then 2nd part of statement 4 also must be false leaving 1st part to be true. That means the car was Toyota. But this makes statement 6 entirely false (as we already know color of car is blue). This contradicts the crucial data given - Each of the other five provided one true and one false piece of information. In the case, there will be no eyewitness giving full correct description.

So the entire Statement 2 must be true. Hence, the car was with it's headlight off.

If headlights were turned off then 2nd part of the Statement 4 must be true and 1st part false. That means, car wasn't Toyota.

And if car wasn't Toyota, as per Statement 6, it must be Ford but not of gray color.
This matches our early conclusion where we concludes color of the car was blue.

Conclusions:

1.What was the car’s brand?

   - Ford

2.What color was the car?

   - Blue

3.Was the car going fast or slow?

   - Fast

4.Did it have license plates?

   - Yes, it had.

 

5.Were its headlights turned on?

   - No, those were off.

6.Was the driver a man or a woman?

   - A woman. 

The Monty Hall Problem

You’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?”

Is it to your advantage to switch your choice?

Will you switch or stay with your door?

Note : Monty Hall was the host of game show called 'Let's Make a Deal' on which above puzzle is based. (Source-Wikipedia)



This is what you should do! 

Winning The Monty Hall Game Show

What was the game show?

Suppose you always choose DOOR 1. Then host will open DOOR 2 or DOOR 3 behind which car is not there.

If the car is behind the DOOR 1, then host will open the DOOR 2 or DOOR 3. And if you switch to remaining DOOR 3 or DOOR 2, you will find goat behind it & you will loose.

And if the car is behind the DOOR 2, then host will be forced to open DOOR 3. Now, if you switch your choice to DOOR 2 then you will win the car behind that door.

Again, if the car is behind the DOOR 3, then host has to open the DOOR 2 behind which goat is there. And now if you switch your selection from DOOR 1 to DOOR 3, then you will be winning the car.

So out of 3 possibilities, in 2 you will be winning this game show if you switch your choice. The probability of winning the game show is 2/3.

And if you stay with your first choice, then probability of having car behind selected door is 1/3.

To conclude, it's better to switch the choice as it increases the probability of winning the game show from 1/3 to 2/3. 
 

Aeroplane Probability Puzzle

People are waiting in line to board a 100-seat airplane. Steve is the first person in the line. He gets on the plane but suddenly can’t remember what his seat number is, so he picks a seat at random. After that, each person who gets on the plane sits in their assigned seat if it’s available, otherwise they will choose an open seat at random to sit in.

 
The flight is full and you are last in line. What is the probability that you get to sit in your assigned seat?

It's 1/2. Shocked? Read how! 

Aeroplane Probability Puzzle - Solution

What was the puzzle?

The probability is really 1/2.

Let's assume that my name is Steve who was the first person who gets on the plane. And you are the person who is last in line. Also, let Seat No.1 was my assigned seat & Seat No.100 is your assigned seat.

Now if I choose seat no.1 itself, rest of all will get their assigned seat including you.

But what if I choose random seat say seat no.15? All the passengers from 2 to 14 will get their assigned seats.

Now, passenger no.15 can choose seat assigned to me i.e. seat no.1 or seat no.100 i.e. your seat. If he sits on my seat i.e. seat no.1 then rest of all including you will get the respective assigned seats. And if he chooses your seat i.e. seat no.100 then all passengers from 16-99 will get their respective assigned seats but you will be left with only option of seat left vacant by me i.e. seat no.1.

The passenger no.15 can choose any random seat as well. In that case, he leaves your 'result' to the passenger whose assigned seat he chooses. Suppose he chooses seat no.76. Again all passengers from 16 to 75 will be getting their respective seats. Now, passenger can choose seat no.1 or seat no.100 or any seat from 77-99.

Again, if he chooses my seat no.1 then rest of all i.e. 77 to 100 which includes you will be getting assigned seats. And if he selects your seat no.100 then again 77-99 will be getting their respective seats but my seat no.1 will be empty for you. However, he can leave your 'result' to any body from 77-99 whose seat he is going to occupy.

In short, your 'result' is in hand of the passenger whose assigned seat is already occupied. He can choose my seat to give you your assigned seat or he can select your seat leaving my seat for you as rest of all passenger will be getting their respective assigned seats. And he can leave your 'result' in hand of other passenger whose seat he randomly occupies.

Suppose, the passenger no.76 above selects seat no.99 in random. Now, all from 77-98 will have no problem in getting assigned seats but passenger 99 will have only 2 options either seat no.1 or seat no.99.

In the end, whenever the passenger ahead of you gets on the plane then either he will get his assigned seat if somebody occupied my seat hence you will get your assigned seat. And if my seat is vacant, then probability that he chooses my seat or your seat is 1/2.

So you will find either my seat or your seat empty when you get on the plane. That is it could be assigned seat or not assigned seat. The probability that the seat you have chosen is assigned must be 1/2.

Prove The Mathematical Fact

Show that the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B.

Here is the proof!