One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $\mathrm{E}$ and $\mathrm{F}$ independent ?
$E:$ 'the card drawn is a spade'
$F:$ 'the card drawn is an ace'
One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $\mathrm{E}$ and $\mathrm{F}$ independent ?
$E:$ 'the card drawn is a spade'
$F:$ 'the card drawn is an ace'
In a deck of $52$ cards, $13$ cards are spades and $4$ cards are aces.
$\therefore $ $ \mathrm{P}(\mathrm{E})=\mathrm{P}$ (the card drawn is a spade) $=\frac{13}{52}=\frac{1}{4}$
$\therefore $ $ \mathrm{P}(\mathrm{F})=\mathrm{P}$ (the card drawn is a ace) $=\frac{4}{52}=\frac{1}{13}$
In the deck of cards, only $1$ card is an ace of spades.
$ \mathrm{P}(\mathrm{EF})=\mathrm{P}$ (the card drawn is spade and an ace) $=\frac {1}{52}$
$\mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\frac{1}{4} \frac{1}{13}=\frac{1}{52}=\mathrm{P}(\mathrm{EF})$
$\Rightarrow \mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\mathrm{P}(\mathrm{EF})$
Therefore, the events $\mathrm{E}$ and $\mathrm{F}$ are independent.
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From the employees of a company, $5$ persons are selected to represent them in the managing committee of the company. Particulars of five persons are as follows :
S.No.
Name
Sex
Age in years
$1.$
Harish
$M$
$30$
$2.$
Rohan
$M$
$33$
$3.$
Sheetal
$F$
$46$
$4.$
Alis
$F$
$28$
$5.$
Salim
$M$
$41$
A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over $35$ years?
From the employees of a company, $5$ persons are selected to represent them in the managing committee of the company. Particulars of five persons are as follows :
| S.No. | Name | Sex | Age in years |
| $1.$ | Harish | $M$ | $30$ |
| $2.$ | Rohan | $M$ | $33$ |
| $3.$ | Sheetal | $F$ | $46$ |
| $4.$ | Alis | $F$ | $28$ |
| $5.$ | Salim | $M$ | $41$ |
A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over $35$ years?
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True statement $A$ and true statement $B$ are two independent events of an experiment.Let $P\left( A \right) = 0.3$ , $P\left( {A \vee B} \right) = 0.8$ then $P\left( {A \to B} \right)$ is (where $P(X)$ denotes probability that statement $X$ is true statement)