Shuffle Play You use a shuffle playback feature to randomly pl...

Question

Shuffle Play You use a shuffle playback feature to randomly play songs in a playlist. The playlist of 56 songs includes 15 instrumental songs.

a. What is the probability that the first three songs to play are instrumental songs? (Assume a song cannot be repeated.)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. You have a playlist of 40 songs, 9 of which are rock songs. If you play 3 songs at random in succession without repeats, what is the probability that all 3 songs played are rock songs? Awesome. So it appears for this particular prompt, we're ultimately trying to determine what is the probability that all three songs played are rock songs if we were to play 3 songs at random in succession, meaning one after the other without any repeats, what is our probability that all 3 of those randomly chosen songs are rock songs specifically? So with that in mind, now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be, noting that they're all stating that they're about some value. So A is 0.0021, B is 0.0085, C is 0.0129, and finally D is 0.0156. Awesome. So our first step in order to solve this particular problem is we need to find the number of favorable outcomes, meaning all three are rock songs and that the order in this particular case for this prom does indeed matter. So therefore, with that in mind, as we should recall a note, the number of ways to select and arrange 3 rock songs out of 9 is the probability of 9.3 or 9 chooses 3. is equal to 9 factoral divided by parentheses 9 minus 3 multiplied by factoral. And when we plug that into our calculator, we will find that this is equal to 504. Fantastic. And our second step is we need to find the total number of possible outcomes, which once again is any 3 songs where the order in this particular case does indeed matter. So as we should recall and note, the total number of ways to select and arrange three songs from 40 is P 40.3 or 40 chooses 3 is equal to 40 factoral divided by 40 minus 3 multiplied by factoral, which when we plug it into our calculator. We will find that this is equal to 59,280. Fantastic. So, our third and final step is we need to determine the probability. So the probability P is equal to P 9.3 divided by P of 40.3, which is equal to when we plug in our known values, 504 divided by 59,280, which when we plug that into our calculator and we round to 4 decimal places to match all of our multiple choice answers, that will mean that our final answer is approximately equal to 0.00. 85. So that will mean our final answer, without a doubt, looking at our multiple choice answers will have to be multiple choice answer letter B, which is about 0.0085. So therefore, without a doubt, the probability that all three songs played in succession are being rock songs is about 0.0085. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Shuffle Play You use a shuffle playback feature to randomly play songs in a playlist. The playlist of 56 songs includes 15 instrumental songs.
a. What is the probability that the first three songs to play are instrumental songs? (Assume a song cannot be repeated.)

Key Concepts

Probability

Combinatorial Counting

Dependent Events

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