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A standard deck of playing cards (with jokers removed) consists of $52$cards in four suits鈥攃lubs, diamonds, hearts, and spades. Each suit has $13$cards, with denominations ace,$2,3,4,5,6,7,8,9,10$jack, queen, and king. The jacks, queens, and kings are referred to as 鈥渇ace cards.鈥� Imagine that we shuffle the deck thoroughly and deal one card. The two-way table summarizes the sample space for this chance process based on whether or not the card is a face card and whether or not the card is a heart.

Are the events 鈥渉eart鈥� and 鈥渇ace card鈥� independent? Justify your answer.

The Pew Research Center asked a random sample of $2024$adult cell-phone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data.

Suppose we select one of the survey respondents at random.

a. Find $P\left(iPhone\right|18-34)$.

b. Use your answer from part (a) to help determine if the events 鈥渋Phone鈥� and $"18-34"$are independent.

Cell phones The Pew Research Center asked a random sample of $2024$adult cell-phone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data.

Suppose we select one of the survey respondents at random.

1. Find P(iPhone | $18-34$)
2. Use your answer from part (a) to help determine if the events 鈥渋Phone鈥� and $"18-34"$are independent.

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.

Suppose we select one of these students at random.

a. Find $P\left(athletic\right|5thgrade)$.

b. Use your answer from part (a) to help determine if the events 鈥�$5th$ grade鈥� and 鈥渁thletic鈥� are independent.

Middle school values Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.

Suppose we select one of these students at random.

1. Find P(athletic | $5^{th}$grade).
2. Use your answer from part (a) to help determine if the events 鈥�$5^{th}$ grade鈥� and 鈥渁thletic鈥� are independent.

Rolling dice Suppose you roll two fair, six-sided dice鈥攐ne red and one green. Are the events 鈥渟um is 7鈥� and 鈥済reen die shows a 4鈥� independent? Justify your answer. (See Figure 5.2 on page 314 for the sample space of this chance process.)

Rolling dice Suppose you roll two fair, six-sided dice鈥攐ne red and one green. Are the events 鈥渟um is 8鈥� and 鈥済reen die shows a 4鈥� independent? Justify your answer. (See Figure 5.2 on page 314 for the sample space of this chance process.)

Free downloads? Illegal music downloading is a big problem: 29% of Internet users download music files, and 67% of downloaders say they don鈥檛 care if the music is copyrighted. 18 Find the probability that a randomly selected Internet user downloads music and doesn鈥檛 care if it鈥檚 copyrighted.

. At the gym Suppose that 10% of adults belong to health clubs, and 40% of these health club members go to the club at least twice a week. Find the probability that a randomly selected adult belongs to a health club and goes there at least twice a week.

Box of chocolates According to Forrest Gump, 鈥淟ife is like a box of chocolates. You never know what you鈥檙e gonna get.鈥� Suppose a candymaker offers a special 鈥淕ump box鈥� with 20 chocolate candies that look alike. In fact, 14 of the candies have soft centers and 6 have hard centers. Suppose you choose 3 of the candies from a Gump box at random. Find the probability that all three candies have soft centers