Sarvesh K Verma Solutions for Chapter: Probability, Exercise 5: CAT - Test Questions Helping You Bell the CAT

The probabilities that a student passes in Mathematics, Physics and Chemistry are $\mathrm{m}, \mathrm{p}$ and $\mathrm{c},$ respectively. Of these subjects the student has $75\%$ chance of passing in at least one, a $50\%$ chance of passing in at least two and a $40\%$ chance of passing in exactly, two, which of the following relations are true.

A student appears for tests $A,B$ and $C$. The student is successful if he passes either in tests $A$ and $B$ or tests $A$ and $C$. The probabilities of the student passing in tests $A,B,C$ are $p,q$ and $\frac{1}{2}$ respectively. If the probability that the student is successful is $\frac{1}{2}$ then,

If $\frac{1+4\mathrm{p}}{\mathrm{p}},\frac{1-\mathrm{p}}{4},\frac{1-2\mathrm{p}}{2}$ are probabilities of three mutually exclusive events then

A man can take a step forward, backward, left or right with equal probability. Find the probability that after nine steps he will be just one step away from his initial position.

The digits $1, 2, 3, \dots , 9$ are written in random order to form a nine digit number. Find the probability that this number is divisible by $11.$

A terror outfit, works under the pseudonym of Mafia Wars, uses $12$ satellite phones, during its operation to communicate with each other. Each of the $12$ terrorists is armed with a satellite phone, an $\mathrm{AK}-47$ and hand grenades among other explosives. Except the four terrorists, who are hidden in a tiny staff washroom of a prominent hotel, no two terrorists are positioned at the same location. The terrorists who are hidden in the washroom are codenamed as Alpha and the ones who are hidden at other surrounding locations are codenamed as Beta. Due to location differences Beta terrorists cannot communicate without the satellite phone, at all. In order to speak to any of the Alpha terrorist a Beta terrorist can call on any one of the phones available with Beta, similarly to speak to any Beta terrorist an Alpha can call from any one of the phones available with them. A conference call can be made between any three satellite phones. If three phones are chosen randomly from the $12$ phones, then what is the probability that the terrorists can make a conference call?

Zuckerberg, my Facebook friend, recently returned from his honeymoon trip to three European cities - Paris, Milan and Zurich, where he had clicked some photos before uploading them on Facebook in three different folders naming them on the cities he had visited. The folders Paris, Milan and Zurich have $3, 4,$ and $5$ photos, respectively. He asks his six-year-old niece Olivia Bee that if she could download these photos and edit them using Instagram and put them back but only one photo in each folder. Within no time she edits all the photos and uploads back quickly one photo in each folder. What's the probability that she uploads at least two photos of the same city and no folder has the photo uploaded back to its original folder, after editing?

In a regular decagon, there are diagonals of distinct sizes. If all the possible diagonals are drawn and you choose any one diagonal at random then what is the probability that it is neither the shortest one nor it is the longest one?