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Assume that adults with smartphones are randomly selected in meetings and classes. Find the probability of them using smartphones in classes or meetings.

This question aims to find the probability of adults using smartphones in meetings or classes when phone users are randomly selected.

One of the largest smartphone manufacturers LG surveyed smartphone usage among adults in the social environment like meetings and classes and it was found that 54% of the adults use smartphones in meetings and classes.

Assuming a certain number of smartphone users are selected randomly, we can find the probability of these users using smartphones. If we select 8 adult smartphone users randomly in meetings or classes, we can easily find the probability of 6 smartphone users. 

Probability is defined as the number of chances in which an event can occur randomly. It gives the possible outcomes of the occurrence of an event.

There is various kind of probabilities. Some of them are theoretical probability, experimental probability, and axiomatic probability.

Expert Answer

The given data is as follows:

\[ p = 54 % \]

\[ p = \frac { 54 } { 100 } = 0 . 54 \]

\[ n = 8 \]

Where p is the percentage of smartphone users and n is the total number of randomly selected users.

Binomial probability is the type of probability that takes two outcomes of an event. One of the two outcomes is success which is more likely expected while the other outcome is a failure.

The formula of binomial probability is:

\[   P  (  X  =  x  )  =  \frac {  n  !  }  {  x  !  (  n  –  x  )  ! } . p  ^  x. (  1  –  p )  ^  {  n  –  x }  \]

By putting values in the formula:

\[  P  (  X  =  6 ) =  \frac  { 8 ! } {  6 ! (  8  –  6  )  !  } . 0 . 54  ^  6 . (  1  –  0 . 54  ) ^  {  8  –  6  }   \]

\[  P  (  X  =  6 ) =  \frac  { 8 ! } {  6 ! (  2  )  !  } . 0 . 54  ^  6 . (  1  –  0 . 54  ) ^  {  2 }   \]

\[  P  (  X  =  6 ) =  28 . 0 . 54  ^  6 . 0  .  46  ^  2   \]

\[  P  (  X  =  6 )  \approx 0 . 1469  \]

Numerical Solution

The probability of adults using smartphones in meetings or classes is approximately $ 0.1469 % $.

Example

Samsung surveyed the users of smartphones and found that 44% of adults use smartphones in social gatherings. Find the probability of 6 adult users out of 8 randomly selected users.

\[   P   (  X   =   6 )  =   \frac {  8 !  }  {  6 !  (  8  –  6 )  !  } . 0 . 44  ^  6 . (  1  –  0 . 44 )  ^  {  8  –  6  }   \]

\[  P  (  X  =  6 ) = 28 . 0 . 44  ^  6 . 0  .  56  ^  2  \]

\[  P  (  X  =  6 ) \approx  0 . 0637 \]

The probability of Samsung users out of 8 users is $ 0. 637 % $

Image/Mathematical drawings are created in Geogebra.

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