Probability is a fundamental concept in mathematics, used to measure the likelihood of events occurring. It plays a crucial role in various fields, including statistics, finance, and science, helping us make informed decisions based on uncertain outcomes.
Probability Scale
- 0 (Impossible): Event cannot happen.
- 1 (Certain): Event will definitely happen.
- Values between 0 and 1 represent varying levels of likelihood.
![Probability Scale](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F22069d3a-d637-450c-95d5-e3709bb09973-file.png&w=1080&q=75)
Probability Notation
- P(A): Probability of event A.
Worked Examples
Example 1: Flipping a Coin
Consider rolling a standard six-sided dice.
- Success event: Rolling a four.
- Total possible outcomes: 6 (one for each side of the dice).
Calculation:
P(Four)=Total possible outcomesNumber of ways to achieve a four=61This shows that the probability of rolling a four is approximately 0.167, indicating it is less likely to occur compared to flipping heads on a coin.
![Flipping of Coin](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fd5641f37-4f6b-4ba7-ae2e-5db57f34a1ac-file.png&w=1080&q=75)
Image courtesy of Pixabay
Example 2: Rolling a Dice
Consider rolling a standard six-sided dice.
- Success event: Rolling a four.
- Total possible outcomes: 6 (one for each side of the dice).
Calculation:
$
P(\text{Four}) = \frac{\text{Number of ways to achieve a four}}{\text{Total possible outcomes}} = \frac{1}{6}
<p></p><p>Thisshowsthattheprobabilityofrollingafourisapproximately0.167,indicatingitislesslikelytooccurcomparedtoflippingheadsonacoin.</p><imgsrc="https://tutorchase−production.s3.eu−west−2.amazonaws.com/2658ce21−2049−42ff−b388−ee917fa999de−file.png"alt="Dice"style="width:500px;height:465px"width="500"height="465"><h2id="understanding−zero−and−one−in−probability"><strong>UnderstandingZeroandOneinProbability</strong></h2><ul><li><strong>Probabilityof0:</strong>Consideraneventlikedrawingaredcardfromastandarddeckof52bluecards.Sincetherearenoredcards,theprobabilityis0,makingitanimpossibleevent.</li></ul><p></p><ul><li><strong>Probabilityof1:</strong>Ifyouweretodrawacardfromadeckcontainingallredcards,theprobabilityofdrawingaredcardwouldbe1,indicatingcertainty.</li></ul><h3><strong>Example1:DrawingfromaDeckofBlueCards</strong></h3><ul><li><strong>Event</strong>:Drawingaredcardfrombluecards.</li><li><strong>Probability</strong>:0(Impossible).</li></ul><imgsrc="https://tutorchase−production.s3.eu−west−2.amazonaws.com/8e4e37bc−f603−4d1a−804f−d58bb66e4e04−file.png"alt="BlueDeckofCards"style="width:500px;height:285px"width="500"height="285"><h3><strong>Example2:DrawingfromaDeckofRedCards</strong></h3><ul><li><strong>Event</strong>:Drawingaredcard.</li><li><strong>Probability</strong>:1(Certain).</li></ul><imgsrc="https://tutorchase−production.s3.eu−west−2.amazonaws.com/20fad189−43ea−4687−9b1b−01be637cc037−file.png"alt="RedCards"style="width:500px;height:505px"width="500"height="505"><h2id="visualising−probability"><strong>VisualisingProbability</strong></h2><ul><li>Drawinganacefromadeckofcards.</li><li><strong>Totalnumberofacesinadeck</strong>:4.</li><li><strong>Totalcardsinadeck</strong>:52.</li></ul><p></p>P(\text{Ace}) = \frac{4}{52} = \frac{1}{13}
<p></p><p>Plottingthisontheprobabilityscale,weseethatP(Ace)iscloserto0thanto1,indicatingarelativelylowprobability.</p><h2id="real−world−applications"><strong>Real−WorldApplications</strong></h2><p>Probabilityscaleisnotjustatheoreticalconcept;ithasreal−worldapplicationsin:</p><ul><li><strong>Decisionmaking</strong>:Helpsinassessingrisksandmakinginformeddecisions.</li><li><strong>Predictivemodels</strong>:Usedinweatherforecasting,stockmarketanalysis,andmore.</li></ul><h2id="practice−questions">PracticeQuestions</h2><h3><strong>Question1</strong></h3><p>Whatistheprobabilityofdrawingaqueenfromastandarddeckofcards?</p><p></p><h4><strong>Solution:</strong></h4><ul><li><strong>Totalnumberofqueensinadeck</strong>:4.</li><li><strong>Totalcardsinadeck</strong>:52.</li></ul><p></p>P(\text{Queen}) = \frac{4}{52} = \frac{1}{13}
<p></p><h3><strong>Question2</strong></h3><p>Ifabagcontains3redmarblesand7bluemarbles,whatistheprobabilityofdrawingaredmarble?</p><p></p><h4><strong>Solution:</strong></h4><p></p>P(\text{Red Marble}) = \frac{3}{10}$