# A card is selected at random from a well-shuffled pack of $52$ cards.

Find the probability that the selected card is

$(1)$A black coloured queen

$(2)$not a king

Last updated date: 26th Mar 2023

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Answer

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Hint- Probability is the ratio of favourable events to total number of events. Probability of not a king \[P\left( {B'} \right) = 1 - P\left( B \right)\].

As it is given $52$ cards well shuffled. so, for a given experiment the number of equally likely outcomes we take as \[n = 52\].

$(1)$Now let A be the event that the selected card is a black coloured queen and we know that in a deck of $52$ cards only $2$ black coloured queen cards are there i.e. queen of spade or club.

The number of favourable outcomes is $2$.

\[\therefore P\left( A \right) = \frac{2}{{52}} = \frac{1}{{26}}\]

$(2)$Now for this case let B be the event that the selected card is a king. The number of favourable outcomes is$4$as we know that in a deck of $52$ cards only $4$ black king cards are there.

\[\therefore P\left( B \right) = \frac{4}{{52}} = \frac{1}{{13}}\]

Then, let ${\text{B'}}$be the event that the selected card is not a king.

\[\therefore P\left( {B'} \right) = 1 - P\left( B \right) = 1 - \frac{1}{{13}} = \frac{{12}}{{13}}\]

Hence the answer is \[\frac{{12}}{{13}}\].

Note: Probability questions are based on the number of outcomes by total outcomes. Probability can neither be negative nor be greater than $1$.A deck has $52$ cards having a set of diamonds, clubs, hearts and spades each having $13$ cards. $P\left( A \right)$is a short way to write probability of event A.

As it is given $52$ cards well shuffled. so, for a given experiment the number of equally likely outcomes we take as \[n = 52\].

$(1)$Now let A be the event that the selected card is a black coloured queen and we know that in a deck of $52$ cards only $2$ black coloured queen cards are there i.e. queen of spade or club.

The number of favourable outcomes is $2$.

\[\therefore P\left( A \right) = \frac{2}{{52}} = \frac{1}{{26}}\]

$(2)$Now for this case let B be the event that the selected card is a king. The number of favourable outcomes is$4$as we know that in a deck of $52$ cards only $4$ black king cards are there.

\[\therefore P\left( B \right) = \frac{4}{{52}} = \frac{1}{{13}}\]

Then, let ${\text{B'}}$be the event that the selected card is not a king.

\[\therefore P\left( {B'} \right) = 1 - P\left( B \right) = 1 - \frac{1}{{13}} = \frac{{12}}{{13}}\]

Hence the answer is \[\frac{{12}}{{13}}\].

Note: Probability questions are based on the number of outcomes by total outcomes. Probability can neither be negative nor be greater than $1$.A deck has $52$ cards having a set of diamonds, clubs, hearts and spades each having $13$ cards. $P\left( A \right)$is a short way to write probability of event A.

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