Mathematics, 08.12.2020 16:50 bryantjennifer1966

The Cat on the Piano A standard piano keyboard has 88 different keys. Find the probability that a cat, jumping on 4 keys in sequence and at random (possibly with repetition), will strike the first four notes of Beethoven's Fifth Symphony. (Leave your answer as a formula.)

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The probability that a cat, jumping on 4 keys in sequence and at random, will strike the first four notes of Beethoven's Fifth Symphony is $\rm 1.7\times 10^{-8}$ .

Given :

The Cat on the Piano A standard piano keyboard has 88 different keys.

The following steps can be used in order to determine the probability that a cat, jumping on 4 keys in sequence and at random, will strike the first four notes of Beethoven's Fifth Symphony:

Step 1 - The four notes of the piano are 3 LA's and 1 REb.

Step 2 - The probability that the first key is LA is 1/88.

Step 3 - The probability that the second key is LA is 1/88.

Step 4 - The probability that the third key is LA is 1/88.

Step 5 - The probability that the key is REb is 1/88.

Step 6 - So, the probability that a cat, jumping on 4 keys in sequence and at random, will strike the first four notes of Beethoven's Fifth Symphony is:

$\rm P=\dfrac{1}{88}\times \dfrac{1}{88}\times \dfrac{1}{88}\times \dfrac{1}{88}$

$\rm P = 1.7\times 10^{-8}$

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Answer from: miracle96

The piano has 88 keys, so there are 88 keys that can be pressed by the cat.

We can assume that each one of the 88 keys has the same probability of being pressed.

We want to find the probabiiity that the cat press the first four notes of the Fifth Symphony (i suppose we want to find the probability where the cat strikes the notes in the correct order), those notes are:

These notes, at least in the melody, are 3 LA´s, and one REb.

While in a piano the notes are repeated a lot of times, (in 88 keys we will have around 7 of each).

All of them have a different pitch (some are higher notes, and other are lower).

The ones in the beginning of the Fifth Symphony are already defined, this means that for each note, we have a probability of 1/88 of hitting the correct key.

Then the cat needs to do this four times.

The first LA will have a probability of 1/88 of being hit.

The second LA will have a probability of 1/88 of being hit.

The third LA will have a probability of 1/88 of being hit.

The REb will have a probability of 1/88 of being hit.

Then the joint probability will be the product of each individual probability, this means that the probability of the cat hitting those four notes in order is:

P = (1/88)*(1/88)*(1/88)*(1/88) = 1.7*10^(-8)

Answer from: Quest

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