<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Two events are independent if the following are true:</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">&bull; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">A</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">) = </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">A</span></em><span style="color:black">)</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">&bull; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">|</span><em><span style="color:black">A</span></em><span style="color:black">) = </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">)</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">&bull; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">A </span></em><span style="color:black">AND </span><em><span style="color:black">B</span></em><span style="color:black">) = </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">A</span></em><span style="color:black">)</span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">)</span></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">Two events&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>independent events</strong>&nbsp;if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show&nbsp;<strong>only one</strong>&nbsp;of the above conditions. If two events are&nbsp;<strong>not</strong>&nbsp;independent, then we say that they are&nbsp;<strong>dependent events</strong>.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">Sampling may be done&nbsp;<strong>with</strong>&nbsp;<strong>replacement</strong>&nbsp;or&nbsp;<strong>without replacement</strong>.</span></span></span></p>

<p>&nbsp;</p>

<ul>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242"><strong><span style="color:#0070c0">With replacement</span></strong>: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.</span></span></span></p>
	</li>
</ul>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">???????</span></span><span style="color:#424242">A bag contains four blue and three white marbles. James draws one marble from the bag at random, records the color, and replaces the marble. The probability of drawing blue is&nbsp;<span class="math-tex">\(\frac{4}{7}\)</span>. When James draws a marble from the bag a second time, the probability of drawing blue is still&nbsp;<span class="math-tex">\(\frac{4}{7}\)</span>.</span><span style="background-color:white"><span style="color:#424242">James replaced the marble after the first draw, so there are still four blue and three white marbles.</span> </span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238617671623861767.jpg" style="height:501px; width:400px" /></span></p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="background-color:white"><span style="color:#424242">Figure&nbsp;3.7</span></span></strong></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">&nbsp;<strong><span style="color:#424242">Without replacement</span></strong><span style="color:#424242">: </span><span style="color:black">When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The bag still contains four blue and three white marbles. Maria draws one marble from the bag at random, records the color, and sets the marble aside. The probability of drawing blue on the first draw is&nbsp;</span><span style="color:#424242"><span class="math-tex">\(\frac{4}{7}\)</span></span><span style="color:black">. Suppose Maria draws a blue marble and sets it aside. When she draws a marble from the bag a second time, there are now&nbsp;<em>three</em>&nbsp;blue and three white marbles. So, the probability of drawing blue is now&nbsp;</span><span style="color:#424242"><span class="math-tex">\(\frac{3}{6}=\frac{1}{2}\)</span></span><span style="background-color:white"><span style="color:black">. Removing the first marble without replacing it influences the probabilities on the second draw.</span> <span style="background-color:white"><span style="color:black">If it is not known whether&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are independent or dependent,&nbsp;<strong>assume they are dependent until you can show otherwise</strong>.</span></span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="background-color:white"><span style="color:black"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238619131623861913.png" style="height:359px; width:400px" /></span></span><span style="color:black">Fig 3.8</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px"><strong>EXAMPLE&nbsp;3.4</strong></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. Clubs and spades are black, while diamonds and hearts are red cards. There are 13 cards in each suit consisting of&nbsp;<em>A</em>&nbsp;(ace), 2, 3, 4, 5, 6, 7, 8, 9, 10,&nbsp;<em>J</em>&nbsp;(jack),&nbsp;<em>Q</em>&nbsp;(queen),&nbsp;<em>K</em>&nbsp;(king) of that suit.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238619731623861973.jpg" style="height:363px; width:800px" /></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><strong>Figure&nbsp;3.9</strong></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><strong>Sampling with replacement</strong></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Suppose you pick three cards with replacement. The first card you pick out of the 52 cards is the&nbsp;<em>Q</em>&nbsp;of spades. You put this card back, reshuffle the cards and pick a second card from the 52-card deck. It is the 10 of clubs. You put this card back, reshuffle the cards and pick a third card from the 52-card deck. This time, the card is the&nbsp;<em>Q</em>&nbsp;of spades again. Your picks are {<em>Q</em>&nbsp;of spades, 10 of clubs,&nbsp;<em>Q</em>&nbsp;of spades}. You have picked the&nbsp;<em>Q</em>&nbsp;of spades twice. You pick each card from the 52-card deck.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px"><strong>Sampling without replacement</strong></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Suppose you pick three cards without replacement. The first card you pick out of the 52 cards is the&nbsp;<em>K</em>&nbsp;of hearts. You put this card aside and pick the second card from the 51 cards remaining in the deck. It is the three of diamonds. You put this card aside and pick the third card from the remaining 50 cards in the deck. The third card is the&nbsp;<em>J</em>&nbsp;of spades. Your picks are {<em>K</em>&nbsp;of hearts, three of diamonds,&nbsp;<em>J</em>&nbsp;of spades}. Because you have picked the cards without replacement, you cannot pick the same card twice.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;3.5</strong></span></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,&nbsp;<em>J</em>&nbsp;(jack),&nbsp;<em>Q</em>&nbsp;(queen), and&nbsp;<em>K</em>&nbsp;(king) of that suit.&nbsp;<em>S</em>&nbsp;= spades,&nbsp;<em>H</em>&nbsp;= Hearts,&nbsp;<em>D</em>&nbsp;= Diamonds,&nbsp;<em>C</em>&nbsp;= Clubs.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<ol style="list-style-type:lower-alpha">
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:black">Suppose you pick four cards, but do not put any cards back into the deck. Your cards are&nbsp;<em>QS</em>, 1<em>D</em>, 1<em>C</em>,&nbsp;<em>QD</em>.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:black">Suppose you pick four cards and put each card back before you pick the next card. Your cards are&nbsp;<em>KH</em>, 7<em>D</em>, 6<em>D</em>,&nbsp;<em>KH</em>.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:black">Which of a. or b. did you sample with replacement and which did you sample without replacement?</span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

<p><span style="font-size:14px"><strong>Solution&nbsp;3.5</strong></span></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">a. Because you do not put any cards back, the deck changes after each draw. These events are dependent, and this is sampling without replacement; b. Because you put each card back before picking the next one, the deck never changes. These events are independent, so this is sampling with replacement.</span></span></p>