Ivan Mrázek began his athletic career as a swimmer, but switched to basketball in 1944 when he was noticed for his height. It was not long before he found considerable success in the sport, as he was with the Czechoslovakian team when they won EuroBasket in 1946. He would go on to win three silver medals with the national squad at that tournament in 1947, 1951, and 1955, as well as participate in two editions of the Olympics, in 1948 and 1952, where he placed seventh and joint-ninth respectively. At the 1951 EuroBasket, he was the top scorer and MVP. Domestically, he won six national league titles with Spartak ZJŠ Brno (now BC Brno).
Following his retirement from active competition, Mrázek took up coaching and led Spartak ZJŠ Brno to another six national league championships. He also coached the Czechoslovakian team to a fifth-place finish at the 1960 Rome Olympics. In 1991 he was listed among FIBA’s 50 Greatest Players and in 2001 was listed as the third-best Czech basketball player of the 20th century (behind Jiří Zídek and Kamil Brabenec).