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Probability of Independent Events Formula,
Independent Probability Examples

The probability of independent events formula is used when we consider the probability of outcomes that are classed as &nbsp”Independent”&nbsp events.

Independent Events

In probability, events are independent of one another if the outcome of one of the events is NOT affected by the outcome of any of the other events.

So if we have &nbsp2&nbsp different events,&nbsp A&nbsp and &nbspB,

The events are independent if the outcome of event &nbspA&nbsp does NOT affect the outcome of event &nbspB,&nbsp and vice versa.

Probability of Independent Events Formula

When attempting to find the probability of Independent events happening together, we multiply each individual probability together.

With notation for probability, we can see the probability of independent events formula in case of two events.

P( AB ) &nbsp=&nbsp P( A ) × P( B )

Example &nbsp &nbsp


We can consider the case of rolling a dice twice.

Both rolls are independent of each other, the result of the first roll doesn’t affect the result of the second roll.

On the first roll of the dice, the probability of landing a &nbsp3&nbsp is &nbsp\bf{\frac{1}{6}}.

On the second roll of the dice, the probability of landing a &nbsp3&nbsp is also &nbsp\bf{\frac{1}{6}}.

So for the probability of landing two &nbsp3‘s&nbsp from two rolls of a standard dice.

P( 33 ) &nbsp=&nbsp P( 3 ) × P( 3 ) &nbsp = &nbsp \bf{\frac{1}{6}} × \bf{\frac{1}{6}} &nbsp=&nbsp \bf{\frac{1}{36}}

It’s not overly likely that two &nbsp3‘s&nbsp would be rolled in a row.

More than 2 rolls of the Dice?

What if we wanted to know the probability of rolling a &nbsp3&nbsp four times in a row instead of two?

We multiply the four probabilities together.

P( Four 3‘s ) &nbsp=&nbsp \bf{\frac{1}{6}} × \bf{\frac{1}{6}} × \bf{\frac{1}{6}} × \bf{\frac{1}{6}} &nbsp=&nbsp \bf{\frac{1}{1296}}

This is extremely unlikely to happen, you’d expect to have to roll a dice over &nbsp1’200&nbsp times to obtain a sequence of four &nbsp3‘s&nbsp in a row.

Rolling five &nbsp3‘s&nbsp is considerably more unlikely to occur. &nbsp \bf{\frac{1}{6}} × \bf{\frac{1}{6}} × \bf{\frac{1}{6}} × \bf{\frac{1}{6}} × \bf{\frac{1}{6}} &nbsp=&nbsp \bf{\frac{1}{7776}}

However the main point to focus on is, even if four &nbsp3‘s&nbsp in a row have already been rolled.

The probability of a &nbsp3&nbsp on the fifth roll is still &nbsp\bf{\frac{1}{6}}.

Because the previous four dice rolls were Independent of the fifth roll, regardless of what the outcome of each roll was.

Independent Probability Examples


If we were to flip a coin &nbsp3&nbsp times in a row.

What is the probability of landing &nbsp3&nbsp headss in a row?

Solution&nbsp &nbsp

P( Three Heads ) &nbsp=&nbsp P( Head ) × P( Head ) × P( Head ) &nbsp = &nbsp \bf{\frac{1}{2}} × \bf{\frac{1}{2}} × \bf{\frac{1}{2}} &nbsp = &nbsp \bf{\frac{1}{8}} &nbsp=&nbsp 0.125


Deck of cards that can show how to use the probability of independent events formula.

Picking a single card from a standard deck of &nbsp52&nbsp cards.

What is the probability of picking out a card that is a Spade, as well as a Face card?

Solution&nbsp &nbsp

Although we are picking just &nbsp1&nbsp card from a &nbsp52&nbsp card deck.

Picking a Spade card, and picking a Face card, are still separate events that are Independent of each other.

So we can use the same approach to find the probability of picking out &nbsp1&nbsp card, that is both a Face card and Spade card.

P( Spade ) &nbsp= &nbsp \bf{\frac{13}{52}} &nbsp &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp &nbsp P( Face card ) &nbsp= &nbsp \bf{\frac{12}{52}} &nbsp= &nbsp \bf{\frac{3}{13}}

P( Spade & Face card ) &nbsp= &nbsp \bf{\frac{13}{52}} × \bf{\frac{3}{13}} &nbsp= &nbsp \bf{\frac{39}{676}} &nbsp = &nbsp 0.058

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