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If Sam, Tim and 10 other people are randomly positioned in a queue, then what is the probability that Sam and Tim will be positioned next to each other?
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If Sam, Tim and 10 other people are randomly positioned in a queue, th...
Probability that Sam and Tim will be positioned next to each other:

To find the probability that Sam and Tim will be positioned next to each other in a queue of 12 people, we need to consider the total number of possible arrangements where they are next to each other and divide it by the total number of possible arrangements of all the people.

Step 1: Identify the total number of possible arrangements:
Since there are 12 people in the queue, the total number of possible arrangements is 12!.

Step 2: Identify the number of arrangements where Sam and Tim are positioned next to each other:
When Sam and Tim are positioned next to each other, we can treat them as a single entity. So, we have 11 entities (10 individual people + Sam and Tim as a pair) to arrange in the queue. The number of possible arrangements of these 11 entities is 11!.

Step 3: Calculate the probability:
The probability of an event occurring is given by the ratio of the number of favorable outcomes to the number of possible outcomes. In this case, the number of favorable outcomes is the number of arrangements where Sam and Tim are positioned next to each other, which is 11!. The number of possible outcomes is the total number of possible arrangements, which is 12!.

Therefore, the probability that Sam and Tim will be positioned next to each other is:

P(Sam and Tim positioned next to each other) = 11! / 12!

Simplifying this expression:

P(Sam and Tim positioned next to each other) = 11! / (12 * 11!)

The 11! terms cancel out, leaving us with:

P(Sam and Tim positioned next to each other) = 1/12

So, the probability is 1/12 or approximately 0.0833, which means there is a 1 in 12 chance that Sam and Tim will be positioned next to each other in the queue.
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Directions:Read the information given below and answer the question that follows.Five players — Jack, Will, Luke, Benjamin and Franklin —participated in an innovative game show on a slot machine. At the beginning of the game, the five players were asked to stand in a queue, one behind the other, facing the same direction. Whenever a dealer called any player to the slot machine, the player who was called by the dealer had to move to the front of the queue (irrespective of whether his get was correct or not) and the others moved back in the queue, without any other rearrangement. For example, if the third player in the queue was called by the dealer, he had to move to the first position and the players who were earlier in the first and second positions moved to the second and third positions, respectively, while the players in the fourth and fifth positions did not move. Also, exactly one player rolled the slot machine in each attempt. The game had a total of six attempts, and every jackpot hit was awarded 10 dollars and when jackpot was not hit, it carried a penalty of 5 dollars. It is known that no two players had the same number of dollars at the end of the game.It is given that every player had same amount of money in their pocket before the start of the game. Further, the following information is known about their positions in the queue at various stages of the game:(i) Jack was the only player to get two attempts consecutively and he did not get the first attempt.(ii) During the fourth attempt, Will moved from a position behind Franklin to a position in front of him and during the fifth attempt, Luke moved from a position behind Jack to a position in front of him.(iii) Benjamin who was initially (i.e. before the first attempt of the gam e) at the beginning of the queue moved to the last position in the queue by the end of the game.(iv) The sum of the dollars received by the five players was zero and the player who rolled the slot machine in the last attempt won the game.(v) The player who rolled the slot machine in the first attempt had the least number of dollars in the game.(vi) Jack was further ahead in the queue at the end of the game than at the beginning.Q.Which of the following attempts rolled the slot machine to definitely hit the jackpot?

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If Sam, Tim and 10 other people are randomly positioned in a queue, then what is the probability that Sam and Tim will be positioned next to each other??
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